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SEMICENTENNIAL PUBLICATIONS 

OF THE 

UNIVERSITY OF CALIFORNIA 




1868-1918 



ELECTRICAL PHENOMENA 

IN 

PARALLEL CONDUCTORS 

VOLUME I 

ELEMENTS OF TRANSMISSION 



BY 

FREDERICK EUGENE PERNOT, Ph.D. 

Assoc. Member A. I.E. E.; Assistant Professor of Electrical 

Engineering, University of California; Captain, 

Signal Reserve Corps, U.S.A. 



FIRST EDITION Ifefffl^O 

IS', 19 21 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

1918 



Copyright, 1918, 

BY 

FREDERICK EUGENE PERNOT 



Stanhope £CC3M 

F. H. GILSON COMPANT 
BOSTON, U.S.A. 



PREFACE 



It is of fundamental importance to be able to predetermine with 
as much numerical accuracy as possible, the phenomena which 
may take place in electrical systems. It is of importance first, 
because such predetermination, when possible, can be made at 
much less expense and trouble than is required to obtain the same 
information experimentally; and second, because information so 
obtained Opens a way to new developments and improvements 
in the application of scientific knowledge. To predetermine in 
a numerical way what may happen in electrical systems under 
different conditions requires that the various phenomena be 
expressed in a mathematical form or the equivalent thereof. Of 
course it is not the mathematical expressions themselves which are 
of main interest, but these expressions are necessary since they 
afford the only means at our disposal, other than experimental 
investigation, by which the desired knowledge may be obtained. 

In this volume an attempt has been made to set forth the 
mathematical developments leading to solutions for a number of 
problems arising in connection with the transmission of electrical 
energy over metallic circuits. The fundamental equations for the 
propagation of currents and voltages along an electrical circuit 
consisting of parallel conductors or the equivalent thereof have 
been developed and presented by a number of writers — Heaviside, 
Fleming, Pupin, Steinmetz, Kennelly and others; but it has been 
the writer's experience to note that the application of these de- 
velopments to present-day engineering problems has been com- 
paratively rare. This has been due partly to the unfamiliarity 
of a great many of engineers with the particular type of mathe- 
matics necessarily involved in such discussions, and partly to the 
fact that the presentation of the material in various publications 
has not been in such a form as particularly to invite its application. 
Instead of developing and applying to problems rigorous methods 
of solution, the tendency has been to develop apparently simple 
approximate methods; but investigation shows that in many cases 



IV PREFACE 

the simplicity is only apparent and that accurate results may be 
obtained with as little mechanical labor as the approximate. 
Further, a realization or understanding of accurate methods leads 
to an ability to develop approximate methods where such are 
needed on account of the complexity of rigorous solutions, and to 
apply these approximate methods with due consideration of their 
limitations and significance. 

For some time the author has felt that it would be of real 
servifie to have collected under one head as much as possible 
of the available material dealing with the present subject, as well 
aa the addition of such further developments as may be possible. 
It seemed desirable, also, to include in this work the discussion of 
a number of matters, which, though not falling directly under the 
above title, are nevertheless used so often in connection with the 
particular problems which do fall under this head that a complete 
understanding thereof is essential. Accordingly, in this volume 
a discussion of periodic and alternating quantities in general, with 
methods of analysis of periodic curves into Fourier's series, has been 
included. A discussion of the more common form of oscillograph, 
methods for its use and an analysis of its mode of operation, was 
deemed desirable. The fact that at the present date the oscillo- 
graph plays such an important part in the study of transmission- 
line phenomena leads to the belief that its discussion here is not 
irrelevant. 

In regard to the treatment of transmission-line phenomena 
proper, this volume is to be thought of as forming an introduction 
to subsequent volumes dealing with specialized forms of electrical 
transmission, and therefore it naturally has for its scope a dis- 
cussion of some of the more general properties of transmission 
systems only. Continuously alternating-current phenomena only 
have been discussed, for the introduction of methods for deter- 
mining the transient currents or voltages in a conductor or system 
of conductors leads to solutions more complicated than seem 
advisable here. Further, desirable forms of such solutions differ 
so greatly for different types of circuits which may arise that no 
one particular form could be considered sufficiently general to be 
treated in this introductory volume. 

Between points of discontinuity in a simple line carrying con- 
tinuously alternating electrical quantities, the currents and volt- 
ages at one point are related to the currents and voltages at 



PREFACE V 

another point by simple linear relations, and these relations are 
naturally expressed in terms of hyperbolic functions. Since in 
alternating-current work complex quantities are used for the 
representation, analytically, of the vectors in a vector diagram, 
our algebraic formulas necessarily involve hyperbolic functions 
of complex variables. This fact has been one of the greatest 
obstacles preventing a more general utilization of rigorous trans- 
mission-line formulas. The portion of this volume which deals 
with transmission-line phenomena may properly be thought of as 
merely an elaboration dealing with the various forms and results 
which may be obtained from the two fundamental linear equations 
relating quantities at one point in a line to similar quantities at 
another point. In many places these linear equations are of 
exactly the same form as the equations for systems of entirely 
different types, — that is, localized circuits, — and therefore our 
discussions, instead of applying to transmission lines only, are 
applicable also to the general electrical circuit for which equations 
of the same nature hold true. The coefficients of such linear 
equations are of course given by different functional relations 
among the various circuit constants when different types of circuits 
are considered. In " Theory and Calculation of Electric Currents " 
by J. L. LaCour and 0. J. Bragstad a discussion of the general 
electrical circuit is given, with reference to the operating charac- 
teristics of circuits carrying uniformly alternating electrical quan- 
tities, which the author has found very illuminating. It is to be 
hoped that further treatments from this standpoint may appear, 
for the generality of the methods employed make such discussions 
of great value. 

Although, in this present volume, some explanation of the 
complex-quantity method of dealing with alternating quantities 
seemed desirable in order to establish for the reader a clear knowl- 
edge of the forms of procedure followed, it has necessarily been 
assumed that those interested would be familiar with the various 
fundamental theorems and ideas concerning the behavior of the 
simple properties which go to make up an electrical circuit. Hav- 
ing clearly in mind the physical significance of the four funda- 
mental constants involved in electrical systems, that is, resistance, 
conductance, self-inductance and electrostatic capacity, and hav- 
ing formulated the methods by which mathematical discussions 
relating to alternating quantities may be carried on, further 



VI PREFACE 

developments may be thought of as only mathematical in their 
nature. The above condition, however, does not obviate the 
necessity of being very familiar, from a physical standpoint, with 
the particular problems in hand, for unless such familiarity exists 
much difficulty will be encountered in carrying through the mathe- 
matical developments leading to solutions which are of interest 
and importance to the practicing engineer and to the physicist. 

The author wishes to express here his appreciation of the 
valuable advice and encouragement offered by Professor Harris J. 
Ryan of Stanford University and of the services rendered by 
Messrs. E. N. D'Oyly and Geo. L. Greves in verifying the various 
mathematical steps and numerical illustrations. 



FREDERICK EUGENE PERNOT. 



Berkeley, California. 
April 27, 1918. 



CONTENTS 



Section Page 

Preface iii 

CHAPTER I 

Phenomena in Continuous Current Transmission over 
Non-Leaky Lines 

1. The Simple Direct Current Circuit 1 

2. Non-uniform Conductors 1 

3. Transmission Phenomena 3 

CHAPTER II 
Direct Current Line with Leakage 

4. General 7 

5. Fundamental Differential Equation and Solution 7 

6. Determination of Integration Constants 8 

7. Solution in Terms of Hyperbolic Functions 9 

8. Particular Solutions for g = and r = 10 

9. Load-end Quantities in Terms of Generator-end Quantities 11 

10. Voltage Regulation 12 

11. Power Relations in General 13 

12. Maximum Power 14 

13. Efficiency 15 

14. Numerical Illustration 17 

15. Effective Resistance of Line and Determination of Constants by 

Measurement 20 

16. Combinations of Leaky Lines 24 

17. Single Generator and Line Supplying Two Loads 25 

18. Single Load Supplied from Two Power Sources 26 

19. Unloaded Line with Double Source of Power Supply 28 

20. Numerical Illustration 29 

21. General Networks « 30 

CHAPTER III 

Periodic and Alternating Quantities. Fourier's Series and 
Analysis of Periodic Curves 

22. General 33 

23. Periodic Quantities and Fourier's Theorem 33 

24. Use of Fourier's Series 33 

vii 



viii CONTENTS 

Section Page 

25. Determination of Unknown Coefficients by Integration 34 

26. Use of the Auxiliary Curves, y' = y cos nx and y" = y sin nx 35 

27. Determination of Coefficients from a Finite Number of Ordinates . . 35 

28. The Cosine Terms 35 

29. The Sine Terms 37 

30. Interpretation of Results — Cosine Terms 38 

31. Interpretation of Results — Sine Terms 39 

32. Particular Orders of Harmonics 39 

33. Limitations to the Use of a Finite Number of Ordinates 40 

34. Summary 41 

35. Mechanical Analysis based upon Integration Processes 42 

36. Mathematical Theory of Analyzer 43 

37. Tables and Forms for Analysis from a Number of Ordinates 45 

38. Selection of the Number of Ordinates. Separation of Harmonics . . 45 

39. Description of Tabular Forms for Complete Analysis 46 

40. Summary 48 

41. Numerical Illustration and Check of Tabular Forms 49 

42. Comparative Accuracy 53 

CHAPTER IV 

Treatment op Non-sinusoidal Alternating Quantities. 
The Use of the Oscillograph 

43. General 60 

44. The Polar Diagram and Representation of Alternating Quantities . . 61 

45. Complex Quantity Representation 63 

46. The Complex Operator — Ratio between Two Alternating Quanti- 

ties Expressed as Vectors , 65 

47. Transformation from Trigonometric Functions of Time to Complex 

Quantity Representation 67 

48. Superposition. Use and Limitations of the Complex Quantity 

Method 67 

49. Numerical Example of Complex Quantity Treatment of Non- 

sinusoidal Alternating Quantities 69 

50. Distorting Circuits for Use with the Oscillograph 73 

51. Oscillographic Determination of Wave Shape of E.M.F 76 

CHAPTER V 

Equations of Motion of the Galvanometer Mirror in the 
Electromagnetic Type of Oscillograph 

52. General , 81 

53. Fundamental Equations of Motion of the Vibrator Strips 82 

54. Determination of Integration Constants — Their Dependence upon 

the Moment of Inertia of the Mirror 83 

55. The Mirror Motion 85 

56. Application of Analysis to a Particular Case 86 



CONTENTS k 

Section Page 

57. The Intensity of the Magnetic Field Surrounding the Strips 90 

58. Electromotive Force Induced in the Vibrator Strips 91 

59. The Free Vibration Frequency of the System when the Inertia of 

the Mirror is Considered 91 

CHAPTER VI 
Alternating Current Phenomena in an Ideal Line 

60. The Ideal Line 95 

61. Fundamental Circuit Constants 95 

62. Fundamental Differential Equations and Solution 96 

63. Auxiliary Circuit Constants 97 

64. Complete Expansion of Exact Solution 98 

65. Interpretation of the Various Terms in Expanded Solution 98 

66. Example of Calculation of the Fundamental Quantities in the 

Expanded Form of Equation 100 

67. Vector Diagrams Representing the Exponential Form of Equations 102 

68. Operating Characteristics from Vector Diagrams 107 

69. Limiting Forms of Vector Diagrams for Infinite Length Line 108 

70. Instantaneous Values from Vector Diagrams 108 

71. Instantaneous Values of Power at Different Points '. Ill 

72. Equations for the Loci of the Ends of the Component Vectors .... Ill 

73. Mechanical Construction of the Equiangular Spirals 113 

74. Introduction of Hyperbolic Functions of Complex Variables 114 

75. Calculation of Constants in Equations for a Line of Fixed Length. . 115 

76. Graphical Treatment for Various Load-end Conditions 117 

77. Power-factor from Vector Diagrams 120 

78. Operating Characteristics of Line with Various Loads 120 

79. Constant Generator Voltage — Rigorous Solution 125 

80. Constant Generator Voltage — Solution by Plotted Curves 127 

81. Infinite Series Form for Transmission Line Solution 133 

82. Illustration of Convergence of Series 134 

83. Line Admittance and Impedance Localized 135 

84. Collection of Formulae for Use in Computing. Tabular Arrangement 138 

CHAPTER VII 

The Propagation Constant V = VZY = a +j0 

85. General 141 

86. Fundamental Equations for a and 141 

87. Wave Length and Velocity of Propagation 142 

88. Values of a, /3, and S when gL = rC 143 

89. Infinite Frequency Values of a, 0, and S 144 

90. Low Frequency Values of a, /3, and S 145 

91. Illustration of Variation of a, /3, and *S, with Frequency, for Power 

Circuit 146 

92. Relations between a, 0, and S i 149 

93. Numerical Illustrations for Telephone Circuit 150 



X CONTENTS 

CHAPTER VIII 

The Equivalent Impedance of Circuits and the Determination 

of the Fundamental Constants by Measurement 

Section Paob 

94. General Expression for Generator-end Impedance 153 

95. Generator-end Impedances under Special Conditions 154 

96. Analogies between Circuits with Distributed and with Lumped 

Constants 156 

97. Limiting Conditions at Infinite Frequencies 160 

98. Infinite-line Impedance 161 

99. Numerical Illustrations of Equivalent Line Impedance 162 

100. Determination of Line Constants from Short-circuit and Open- 

circuit Tests 165 

101. Multiple Solutions 169 

102. Methods of Making Impedance Measurements 169 

103. An Impedance Bridge 170 

104. Precautions in the Use of the Bridge 171 

105. Illustrative Computation 173 

106. Effect of Errors in Original Data 173 

CHAPTER IX 
Change of Wave Shape Along Lines 

107. Method of Treatment of Complex Wave Shapes 177 

108. General Case — Load-end Impedance = Z 177 

109. Short-circuited and Open-circuited Lines 178 

110. Numerical Illustration. 100 Mile Open-circuited Line 181 

111. Equations for Instantaneous Voltage 183 

112. Equations for Instantaneous Currents 184 

113. Discussions of Wave Shapes 185 

114. Dependence of Wave Shapes upon Generator and Load Character- 

istics 188 

115. Double Source of Energy Supply through Terminal Impedances . . 189 

116. Approximations Based upon the Neglect of Line Losses 191 

117. Numerical Comparison of Approximate and Exact Solutions 193 

118. Resonance Frequencies 194 

CHAPTER X 
Power Relations in Long Lines 

119. Instantaneous and Average Power 198 

120. The Separate Components of E and / at any Point 198 

121. The Power Equation 199 

122. Reduction in Form of the Power Equation 200 

123. Final Form of the Power Equation 202 

124. Numerical Example 202 

125. Line Losses 204 



CONTENTS Xi 

Section Page 

126. Unloaded Line 204 

127. Power in Distortionless Lines 204 

128. Load Impedance for a Minimum Line Loss with a given Po 207 

129. Numerical Illustration 209 

130. Line Efficiency 210 

131. Load Power-factor for Maximum Efficiency 211 

132. Numerical Illustration. Variation of Efficiency with Power- 

factor 213 

133. Peculiarities Incident to Maximum-efficiency Transmission 216 

134. Variation of Maximum-efficiency Conditions with Line Length . . . 217 

135. Zero Length Lines 218 

136. Infinite Length Lines 220 

137. General 221 

138. Proof that for Maximum Efficiency the Ratio between e and eo 

Equals the Ratio between i and io, and that the Generator 
Power-factor Angle is Equal and Opposite to the Load Power- 
factor Angle 222 

139. Load Voltage for Maximum All-day Efficiency with Intermittent 

Loads 223 

140. Line Loss with Varying Loads, and the Proper Load Voltage to 

Secure a Minimum of Such Loss 225 

141. Illustrative Example from Daily Load Curve 227 

142. Dependence of Proper Voltage upon the Leakage Coefficient 228 

143. Determination of Resistance and Leakage Losses Separately; 

Graphical Method 229 

144. Illustration of Graphical Method 231 

145. Separate Losses Analytically 232 

146. Illustration of Analytical Method 235 

147. Separate Losses at No Load 235 



CHAPTER XI 

Voltage Regulation 
Case I. Both Components of the Load Become Zero Simultaneously 

148. Preliminary Developments 237 

149. Scalar Values 238 

150. Introduction of General Form of Equations 238 

151. Reactive Volt-amperes Required at Load for Zero Regulation. . . . 239 

152. Conditions for Minimum Voltage Regulation 240 

153. Conditions for any Voltage Regulation 241 

154. Solution in Terms of Load Admittance 242 

155. Numerical Illustration 242 

156. Nature of Curves Relating g and bo 243 

157. Geometrical Construction of Curves 245 

158. Relations between Vector Diagrams and Foregoing Curves 246 

159. General 248 



XU CONTENTS 

Case II. Total Load-end Susceptance Constant 

Section Pagb 

160. Load-end Voltages in Terms of Load-end Admittances 249 

161. Constant Susceptance Required for Given Regulation 250 

162. Real and Imaginary Solutions 250 

163. Numerical Illustration 251 

164. Nature of the Curves 252 

165. Significance of the Coordinates of the Central Point 254 

166. Mechanical Construction of Hyperbolas 254 

Case III. A Portion Only of the Load-end Susceptance Constant 

167. Load-end Voltages 257 

168. Constant Portion of Load-end Susceptance for a Given Regulation . 258 

169. Nature and Location of Curves 259 

170. Mechanical Construction of the Curves 259 

171. Forms of Curves in Special Cases 261 

172. Zero Voltage Regulation 262 

Case IV. Load of Constant Power-factor 

173. Equation Relating Constant Susceptance and Regulation 264 

174. Location of Curves 264 

175. Nature of Curves 265 

176. Mechanical Construction 266 

177. Numerical Illustration 267 

Appendix 

Hyperbolic Functions 269 

Relationships Involving Hyperbolic and Allied Functions 272 

Tables of Logarithms of Hyperbolic Functions 285 

Tabular Forms for Use in the Analysis of Periodic Curves 325 



ELECTEICAL PHENOMENA IN 
PARALLEL CONDUCTORS 



CHAPTER I 



PHENOMENA IN CONTINUOUS CURRENT TRANSMIS- 
SION OVER NON-LEAKY LINES 

1. The Simple Direct Current Circuit. — The simplest con- 
ception of a transmission circuit is that in which an unvarying 
current is .transmitted over a system in which only the ohmic 
resistance of the conductors need be considered. By taking into 
account only the resistance of the conductors, the tacit assumption 
is made that between any two points which constitute the ends of 
the transmission system under discussion (either as actual termini 
or as points at which discontinuities are introduced) there is no 
"leakage" of current from one conductor or "side" of the system 
to the other. The electric current is therefore the same in magni- 
tude at both ends of such a system, while the potential difference, 
or voltage, between the conductors changes from point to point 
because of the electromotive forces consumed by the resistance 
and current flow. 

2. Non-uniform Conductors. — The general case for the above 
is that in which the resistance of the conductors per unit length 
along the system is not constant. Physically, such- a condition 
brings to mind a circuit in which the conductors are of varying 
cross-section or material. 

Since the cross-section or material of the conductor is varying 
from point to point, the resistance of the conductor per unit length 
along the circuit must be defined by the equation 

dR /1 x 

r= dT (1) 



2 CONTINUOUS CURRENT TRANSMISSION 

in which dR is the actual resistance of the circuit included within 
the length dl. This expression can be made to include both wires 
of the system if they are parallel and therefore of equal lengths. 

Let A m cross-sectional area of conductor. 

p «■ specific resistance of conductor material. 
Then 

dR = dR\ -+* dRt — — j — I — 2 — » 

in which the subscripts 1 and 2 refer to the separate wires or 
conductors. Unless otherwise specified, r will be understood to 
be defined by (2), thereby including both conductors. Since r 
varies along the line, in order that the problem be determinate its 
value must be given as a function of distance, from one end of the 
line say, and therefore 

r=fd). 

Obviously, from (2) the total line resistance of the system is 

JrVo Pk 

rdl= I f(l)dl (3) 

«/0 

If E = voltage or potential difference between the conductors 
at any point, the gradient of potential difference at this point is 
defined by 

dE 
F = -jj- = gradient of potential difference. (4) 

If / = current in the conductors and the distance I be counted 
positively from the load end towards the source of energy, 

F = ™ = rl = hf(l). (5) 

The integral of (5) gives the expression for line voltage at any 
point, 

E = Jl f(l)dl = IoR + C. 

R = line resistance between the point in question and the load, 
and therefore the integration constant C is seen to be equal to 
the voltage at the load, E . 

E = E Q + Rh, (6) 



TRANSMISSION PHENOMENA 3 

and at the supply end, where I = lo, 

E = E + R Jo. (7) 

In short, it is seen that even though the resistance be distributed 
in a non-uniform manner, it is entirely unnecessary to consider 
anything except the total value of line resistance, 



Rt= J rdl, 



unless the manner of distribution of the various electrical quanti- 
ties throughout the line length is desired. This simplicity arises 
from the fact that the line current is uniform or constant in value 
throughout the line length, and therefore all integrals reduce to 
expressions involving only the total resistance. Though very 
simple, this matter is one of fundamental importance, for it is due 
to departures from this condition that the more complicated 
solutions become necessary. 

3. Transmission Phenomena. — Utilizing the value of total 
line resistance R t , the entire discussion of such a transmission line 
is nothing more than that of a simple electrical circuit. 

Eq = voltage at load or receiving end. 
7o = current throughout the system. 
Rt = total line resistance. 

Then 

E = E + R Jo = generator voltage. (8) 

Po = Eolo = power at load, in watts. (9) 

P = EIo = Po + Rth 2 = power at generator. (10) 

P' = RJo 2 = total line loss. (11) 

P E 

Eff. ■■ -=• «» -=• ■« efficiency of transmission. (12) 

Mr £j 

The voltage regulation of a transmission line is defined as 

rise in receiving end voltage from load to no load 
load voltage 

the generator voltage being supposed to remain constant after 
throwing off the load. Under load conditions, by equation (8) 
the receiving end voltage is 

Eq = E — RtIo> 



4 CONTINUOUS CURRENT TRANSMISSION 

and at no load, therefore, Eq = E. The rise in voltage from load 
to no load is RJ . 

R I 
Reg- = p — f ~B~T = regulation at load 7 . (13) 

h — tiuo 

As will be noted, equation (13) is not that of a straight line with 7 
as independent variable if the generator voltage E be supposed 
constant for all loads. If, however, the generator voltage be 
adjusted with changing load so that the receiver voltage E remains 
constant, then the regulation will obviously be given by- 
Reg. = -4-^, for constant E . 

With a constant generator voltage E, it is interesting to notice 
the change in the power at the receiver, P , with a change in cur- 
rent I . 

p = EJo = (E- RJ ) h = Eh - RJo 2 . (14) 

From this equation it is seen that P is a quadratic function of I , 
and therefore for any particular value of P there are two possible 
values of current I . Also, for certain values of P the expressions 
for Jo become complex imaginaries, indicating physically that the 
value of P for which the solution was made was larger than the 
line in question is able to transmit at the assumed valjue of genera- 
tor voltage. This is, of course, for positive values of P . 

The maximum amount of power which may be transmitted over 
the line of resistance R t , at a generator voltage E, is to be obtained 
from (14). By differentiation 



thus 



^ = E - 2 RJo = 0, for -maximum P , (15) 

alo 

E 
7o = 17-5- , for maximum P , (16) 

Z lit 



and the maximum power becomes, by substituting (16) in (14), 

(Po)max. = ~ (16a) 

For the current value determined by (16) it is obvious that the 
load resistance is equal to the line resistance, R = Rt, which is the 



TRANSMISSION PHENOMENA 









































i.W 

and 


E 








































i 


Volu 








































130 


260 








































120 


240 


















%/ 


XW\X 




















110 


220 








































100 


200 










vE„ 






























90 


180 






















p / 


















80 


160 






BH. 


















L 


ae/ 


Los 


l 










ro 


140 








































00 


120 








































50 


1D0 








































40 


80 








































80 


60 












P« 




























20 


40 








































10 


20 















































00 



120 



180 



240 



420 



540 



Fig. 1. 



300 360 
I Amperes 
Direct Current Transmission Line Characteristics. Line Resistance, 
0.50 ohm. Generator Voltage Constant, 250 Volts. 





1 




































k.wJ 

and] 








































% 


Volts 


















































































120 


240 




\ 


























E H 










110 


220 






Kej 


































100 


200 








































80 


180 






















_£ff. 


















80 


160 




V 




































70 


140 








































60 


120 








































50 


100 








































40 


80 








































BO 


60 










^ 






























20 


40 






































10 


20 




R phn 


8 








tm i 


l'>ss 





























0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 

Fig. 2. — Direct Current Transmission Line Characteristics. Line Resistance, 
0.50 ohm. Generator Voltage Constant, 250 Volts. 



6 CONTINUOUS CURRENT TRANSMISSION 

familiar relation between internal or line resistance, and load 
resistance, for a maximum power at the receiver. 

Although in the preceding all quantities have been expressed 
in terms of the load current and load voltage, it may be convenient 
to introduce as independent variable the load resistance R . 

Thus, substituting E = R h in equation (12) for line efficiency, 
and simplifying the expression, 

Ko T" tit 

which is an expression independent of either the voltage or current. 
Similarly, in equation (13) for line regulation, 



Reg. - f. 


(18) 


For power, since 




E 

T — inrl T? — T? T 




A o — r, i D ana ejq — nolo, 
Ko "T txt 




E* 

P = EIo = r= — r— ^ = power at generator. 
Ho -r ttt 


(19) 


T> 

Po = Roh 2 = /p , d \ 2 E* = power at load. 


(20) 



For the case of constant generator voltage, Fig. 1 shows a 
number of curves plotted to 7 as independent variable, and Fig. 2 
shows the curves for the same system plotted with the load resist- 
ance R as independent variable. 

The above discussion covers the more important phenomena in 
a simple direct-current transmission where line resistance only need 
be considered. The equations are intended to cover only the case 
of a line supplied with power at one end and loaded at the other. 
Obviously, in the -practical distribution of power the simple con- 
ditions mentioned are not often realized. Such problems as the 
design of a railway feeder to supply power to an interurban rail- 
way, for instance, involve much more than a consideration of the 
simple matters mentioned above. There will be no attempt made 
here to take up such matters as distribution problems, particu- 
larly as there are already a number of publications dealing exclu- 
sively with the practice and economics of power distribution. 



CHAPTER II 
DIRECT-CURRENT LINE WITH LEAKAGE 

4. General. — Under this head will be discussed the case of 
a line whose conductors are of uniform resistance and between 
which there exists a uniformly distributed conductance, or means 
of leakage. A portion of the following has already been presented 
in numerous works, but it is included here for the sake of com- 
pleteness. 

The following discussion is based entirely upon the use of hyper- 
bolic functions, the natural means of expression for the phenomena 
occurring in systems of the character named. In fact, throughout 
the entire discussion of transmission systems, the use of hyperbolic 
functions leads to an elegance of treatment which cannot be at- 
tained by any other means. 

5. Fundamental Differential Equation and Solution. — Let 

r = resistance of both conductors per unit length of line, ohms. 
g = conductance from wire to wire per unit length of line, 

mhos. 
I = distance counted positively towards energy supply. 
E = line voltage at any point at distance, I. 
I = line current at any point at distance, I. 

The resistance of an element of line of length dl is r dl, and the 
conductance between wires is g dl. 
Therefore, taking differentials 



dI = Egdl, dE = Irdl, 

dl „ dE T 

Ji = E9 > Ti =Ir - 



(i) 



Differentiating again with respect to I, 

(2) 



<H_dE 
dP ~ g dl' 
cPE = dl 

7 



dl 2 r dl 



8 DIRECT-CURRENT LINE WITH LEAKAGE 

Substituting the values of the first derivatives from (1) in (2), 



** T O 

w -rgE = 0. 



(3) 



The two equations in (3) are identical in form, and are of the 
general type of linear, second order differential equation 

of which the general integral is 

y = CtP" + C# m *, 
in which mi and m^ are the two roots of the quadratic 

own? + aim -f a* = 0, 

and C\ and C2 are integration constants. 
In (3), the particular case under discussion, 

ao = 1, ai = 0, and 02 = — rg, 
which gives 

m? — rg = 0, 

mi = -r-Vrg, (4) 

mi = —y/rg. 



The solutions of (3) are therefore 



E = A^ 1 + igr^ 



(5) 



where Ai, A 2 , Bi, and B 2 are integration constants to be deter- 
mined by initial conditions. 

6. Determination of Integration Constants. — As initial con- 
ditions, let the current and voltage at the load, or receiving end, 
be designated by 7 and E , respectively. At this end, by the 
arbitrary choice of positive direction along the line 1 = and the 
exponential terms in (5), reduce to unity. 

To evaluate the integration constants, differentiate the first of 



SOLUTIONS IN TERMS OF HYPERBOLIC FUNCTIONS 9 



(5) and substitute the result in the second equation of (1) for the 
point I = 0. Thus, 



dE 
dl. 

Also, by (5), E 



1=0 



Ji=o 



= VrgAi — VrgA 2 = rl 
= A x + A 2 = E Q . 



(6) 



The solution of the two simultaneous equations (6) yields 



(7) 



By symmetry, or by proceeding in an exactly similar manner, 



2 \ * r / 



V?). 

and therefore, in the general integral (5), 

i-J[(*+ji\^.^+(*-i.^)-^ 



(8) 



(9) 



Equation (9) is complete, in that it expresses the voltage and 
current at any point explicitly in terms of the line constants and 
the voltage and current at the load end. In general, it will be 
found that it is most convenient to so utilize the load quantities 
as independent variables, though for some purposes other pro- 
cedures may be more desirable. This matter will be discussed 
later, in connection with the performance of lines carrying alter- 
nating currents. 

7. Solutions in Terms of Hyperbolic Functions. — Although 
complete, equation (9) is not in very convenient form for compu- 
tation. By combining terms, and letting 



v = Vrg, 



(10) 



10 DIRECT-CURRENT LINE WITH LEAKAGE 

a better form results. 



*-|(*«+ «-«)*■ 






(ID 



The parentheses are immediately seen to constitute the two hyper- 
bolic functions, cosh vl and sinh vl, which will be utilized therefor. 
The equations then become 

E = E cosh vl + 7 y - sinh t>Z, 
7 = Jo cosh «2 + #o y - sinh aZ. 



(12) 



v = Vrg may be considered as one of the fundamental line 

constants, as well as the quantity y - or its reciprocal y -. In 

fact, these two quantities completely specify the characteristics 
of a direct-current transmission circuit when operating under 
steady conditions, and as far as computation purposes are con- 
cerned they constitute a more convenient means of specification 
than the simple quantities r and g, though either set is easily 
obtained from the other. 

For future convenience in writing, the line constants will be 
expressed in either one of the alternative forms, as follows: 

r = line resistance per unit length, 
g = leakage conductance per unit length, 
or v = *JTg, 

and * = y^, y = \ = \/l- 



(13) 



The fundamental line equations, on which subsequent develop- 
ments are based, are then 

E = E cosh vl + JoZ sinh vl, 
I = To cosh vl -+- E y sinh vl. 



(14) 



8. Particular Forms of Solutions f or g = and for r = 0. If 

either r or g becomes zero, then equations (12) or (14) assume an 
indeterminate form, which may be evaluated either by the standard 



LOAD-END QUANTITIES 



11 



methods of calculus for the evaluation of indeterminate forms, or 
by a consideration of the actual physical phenomena occasioned 
by one of these constants becoming equal to zero. 
When g = 0, 

E = E + hrl, 

7 = 7 (discussed in Chapter I). 

When r = 0, 

E = Eq 

I = 7 + E gl 



(15) 



9. Load-end Quantities in Terms of Generator-end Quan- 
tities. — It is often desirable to express the quantities E and 7 
in terms of E and 7. It is apparent that such a solution might 
be obtained from (14) by substituting —I for I, which would 
be equivalent to counting distance as positive in the direction of 
energy flow. To leave no doubt as to the validity of such a pro- 
cedure, solve the two equations (14) simultaneously for Eq and 7 . 
The result is 

_, E cosh vl — Iz sinh vl 

tiQ = 



7 = 



cosh 2 vl — sinh 2 vl 

I cosh vl — Ey sinh vl 

cosh 2 vl — sinh 2 vl 



The denominator is seen to be equal to unity, and therefore 

Eq = E cosh vl — Iz sinh vl, 
7 = 7 cosh vl — Ey sinh vl. 



(16) 



Other combinations derived from the original equation (14) are 
of interest and importance. The load at the receiving end may 
be represented by an equivalent resistance, or conductance, 



Ro — 



Eq 



7 ' 

Eq = RqIq, 



Go = p- = 

7o = GqEq, 



Eq 



(17) 



which values when substituted in (14) give 
E = Roh cosh vl + IqZ sinh vl, 
7 = 7 cosh vl + IoRoy sinh vl, 
E = Eq cosh vl + EoGqZ sinh vl, 
I = GqEq cosh vl + E y sinh vl, 



(18) 



12 



DIRECT-CURRENT LINE WITH LEAKAGE 



from which 



£0 = 



Go cosh vl + y sinh vl ' 

E 

Ro cosh vl + z sinh vl 

m E 

cosh vl + GoZ sinh vl ' 
I 



Eq = 
/o = 



cosh vl + R y sinh vl 



(19) 



(20) 



10. Voltage Regulation. — At this point the equations for 
voltage regulation may be developed. The regulation is defined 
in the same way as was done in section 3, Chapter I. 

By equation (14), the generator voltage, which remains constant 
on the removal of the load To, is 

E = E cosh vl + IqZ sinh vl. 

At no load, the conductance Go = 0, and equation (20) serves to 
determine the value of receiver voltage at no load with the voltage 
E at the generator. Let 

E' = no load receiver voltage; 

then by (20) for Go = 

E E cosh vl + IqZ sinh vl 



cosh vl cosh vl 

= E + Ioz tanh vl. 

Reg. = — „ — - = ^ z tanh vl. 

£j0 &Q 



(21) 
(22) 



For any given line, z tanh vl is a constant, and therefore for such 
a condition of operation that the load voltage E is maintained 
constant by adjustment of the generator voltage, the curve of 
voltage regulation when plotted to load current as abscissae will 
be a straight line, the same as for the simple case in which line 
leakage was not considered. The magnitude of the regulation 
will differ. 

The case parallel to that illustrated by equation (13), Chapter I, 



POWER RELATIONS IN GENERAL 13 

where the generator voltage is supposed to remain constant at all 
loads will now be considered. 

E = constant generator voltage, 
7 = load current. 

Solving (14) for the value of E in terms of E and 7 , 

„ E — IoZ sinh vl „ , T , . T .,__. 

Eq = -T-, = E sech vl — Iqz tanh vl. (23) 

By (20) the no-load receiver voltage is E' = E sech vl. 
E' — E IqZ tanh vl 



Reg. = 



Eq E sech vl — Iqz tanh vl 

h 



Ey cosech vl — I 



(24) 



Equation (24), as in the case of equation (13), Chapter I, for the 
line without leakage, is not that of a straight line but of a curve 
passing through the origin and reaching a value of infinity at a 
value of h = Ey cosech vl, which is the value of h occurring when 
the load end of the line is short-circuited. This may be verified 
from equation (19) by placing R = 0. 

11. Power Relations in General. — At the load end of the line 
the power received is 

P = EJo watts. 

At any other point the power is given by 

P = EI, 

E and I being given by (14). Multiplying the two expressions 
in (14), 

P = EI = Eoh cosh 2 vl + Eolo sinh 2 vl 

+ E 2 y sinh vl cosh vl + h 2 z sinh vl cosh vl. (25) 

By the hyperbolic reduction formulae in the Appendix, this reduces 
to 

P = P cosh 2 vl + \ (7 2 z + E 2 y) sinh 2 vl. (26) 

The total line loss is P' = P — Pa. Utilizing this relation in 
(25), 

F = P (cosh 2 vl + sinh 2 vl - 1) 

+ E 2 y sinh vl cosh vl -f I 2 z sinh d cosh vl; 
P' = P 2 sinh 2 vl + {E<?y + h 2 z) sinh vl cosh vl. (27) 



14 DIRECT-CURRENT LINE WITH LEAKAGE 

For lines with very small losses it is desirable to calculate the 
power loss directly by (27) since a much greater accuracy may be 
secured in this way than by calculating P and then numerically 
subtracting P , for in this latter case the difference between two 
large quantities of nearly the same value must be taken. 

In presenting the formulae as developed, it is not expected that 
they are to be applied indiscriminately to any and all cases, for 
though they are rigid, and will therefore yield accurate results 
wherever they are applied, it may not be at all necessary to in- 
troduce the degree of refinement that their use yields. Individual 
cases must be treated individually, if an efficient use is to be made 
of the material at hand. Obviously, there is little sense in calcu- 
lating line phenomena by rigid formulae to a degree of accuracy 
yielded by six-place logarithms, say, when the line constants on 
which the computations are based are perhaps in error by as 
much as two or three per cent. Such matters must be left to 
the judgment of the individual, but they cannot detract from 
the desirability and need of preparing and presenting rigid 
expressions. 

12. Maximum Power. — Following the scheme adopted in 
section 3, Chapter I, the maximum value of power which can be 
transmitted over a leaky line by a generator of constant voltage, 
E, will now be investigated. To do this, E and P will be ex- 
pressed in terms of E and I , and since E is constant the value of 
I which will give a maximum P can be found. 

By (23) E = E sech vl — I^z tanh vl, 

P = E I Q = I E sech vl - I 2 z tanh vl. (28) 

For a maximum, the first derivative of Po with respect to I 
must be equal to zero, and the second derivative must be negative. 

dP 

-rp = E sech vl — 2 IqZ tanh vl = for max. (29) 

cPPn 

-Tj^ = — 2 z tanh vl, which is negative. 

Thus, by (29), the load current for maximum power is 

j 1 = Esechvl = E ^ Q) 

Jp„ = max. 2 z tanh vl 2 z sinh vl 



EFFICIENCY 15 

Substituting (30) in (28) and collecting terms, 

E 2 E 2 

' 4 z sinh vl cosh vl 2zsinh2tfZ 

It is interesting to compare the expression (31) for the maximum 
power which can be transmitted over a leaky line with the expres- 
sion for the maximum power which can be transmitted over a non- 
leaky line, which is, by (16a), Chapter I, 



P 9 max.l =j 



E 2 
4R t 



It is evident that 4 z sinh vl cosh vl = 2 z sinh 2 vl replaces the 
quantity 4 R t when there is leakage present. By introducing into 
(31) the condition that g = 0, the expression should reduce to (32). 
This will be shown to be true. 

Expanding 2 z sinh 2 vl into a series, using the values of z and v 
by (15), 

2 zsinh 2 vl = 2 y -l~2 Vr^jl + ri (2 Vr~gl) z + • • • 1 

= 4 rl + f r 2 gP + • • • . 

Now, all terms except the first contain g, and therefore for g = 0, 
2 z sinh 2 v£ reduces to 4 H, which is identical with the denomina- 
tor of (32), since 

Rt = rl = total line resistance if uniformly distributed. 

13. Efficiency. — In the non-leaky line, by equation (17), 
Chapter I, it is seen that for loads approaching zero (R approach- 
ing infinity) the efficiency approaches unity in value. Such is not 
the case for the leaky line, for it will be shown here that there is a 
definite limit to the possible value of line efficiency, no matter what 
the condition of loading may be. 

Using equation (26), 

Eff. = — = EoI ° 



P Eoh cosh 2 vl + \ {zU + yE 2 ) sinh 2 vl 

(33) 



1/7 V \ 

cosh 2vl-\-~\z-^r-\ry-r) sinh 2 vl 



From (33) it is apparent that the efficiency is in no way depend- 
ent upon the absolute magnitude of either E or 7 but rather 



16 DIRECT-CURRENT LINE WITH LEAKAGE 

upon the ratio of the two, that is, upon R or G . Obviously, also, 

the maximum efficiency will occur for such a value of this ratio as 

will make the term 

Io , Eo 
u = z-^r + y-y = a minimum. 

Letting G = tt, and remembering that y = -, we have 
r>o z 



For a minimum, 
from which 



du • 1 _ 

dG ~ • zG<? ~ ' 

2 2 = 7T5 = Ro 2 , Z = ±/2o. 
<J0 



In the double sign, only the positive can have a physical signifi- 
cance, and by investigating the value of the second derivative it 
can be shown that this sign gives a minimum to the value of u. 
Thus, for a maximum efficiency, 

Eo = IqZ, 
h = E y. 



(34) 



Substituting the ratio of E to 7o as given by (34) in equation 
(33), the maximum efficiency which may be attained is 

cosh 2 vl -\- sinh 2 vl 

The next and final step in this development will be to express 
the load current for which maximum efficiency occurs in terms of 
the generator voltage E, which is supposed to be constant. 

Introducing (23) into (34), for maximum efficiency, 

E sech vl — IqZ tanh vl = zlo, 
from which 

J 1 = E sech vl = -e~ vl (36) 

JES. . max. z (1 + tanh vl) z € 

Comparing (30) and (36) it is seen that the maximum efficiency 
does not occur coincidently with the maximum power at the load 
when the generator voltage is constant. 



NUMERICAL ILLUSTRATION 17 

All of the above formulae involving hyperbolic functions could 
be expanded into series, thereby furnishing approximate solutions, 
but it is not considered at all necessary to include such develop- 
ments here. The above formulae, involving as they do only hyper- 
bolic functions of real or non-imaginary variables, are so con- 
veniently used numerically that to introduce series expressions 
leads to needless complexity. Neither does it appear necessary 
to introduce various approximate methods of solution, such as 
assuming that the entire leakage may be concentrated at the middle 
of the line, or the still closer approximation of placing one-sixth 
of the leakage at each end and two-thirds in the middle of the line. 
The solutions based on such assumptions are approximations, 
which in reality amount to using only a limited number of terms 
in the series expansions of the hyperbolic functions. If at any time 
tables of such functions are not available, it is more convenient 
to introduce the series for the evaluation of the hyperbolic functions 
than to resort to an approximate solution based on the above 
mentioned assumptions. 

14. Numerical Illustration. — Before proceeding further with 
theoretical developments, a numerical illustration of the foregoing 
will be given. 

Consider a line with the following constants per mile of length. 

r = 50.0 ohms. g = 2 X 10~ 4 mhos. 

v = V7g = 0.1000. z = V- = 500.0. y = J g - = 0.00200. 

V g V r 

Let the line length be I = 15.0 miles. 
Then vl = 1.500. 

(a) What voltage would be required at the sending end to give 
a current of 0.050 amperes through a resistance of 200.0 ohms at 
the receiving end? 

Io = 0.050, Ro = 200.0, E = R I = 10.00. 

By (14), E = 10.0 cosh 1.50 + 0.050 X 500 sinh 1.50 

= 10.0 X 2.35241 + 25 X 2.12928 = 76.7561 volts. 

(b) What is the generator current? 

By (14), J = 0.050 cosh 1.50 + 10.0 X 0.00200 sinh 1.50 
= 0.160206 amperes. 



18 



DIRECT-CURRENT LINE WITH LEAKAGE 



(c) What is the voltage regulation under these conditions? 

By (22), Reg. = ^? 500 tanh 1.50 = 2.50 X 0.90515 
1U.U0 

= 2.263 = 226.3 per cent. 

(d) For a constant generator voltage of 100.0 volts, calculations 
of various quantities for different values of load current are given 
in the following table. The equations used are numbered to 
correspond with the text, and are given in the final numerical form 
for the particular case under discussion. 



(23) 

(28) 
(16) 

(21) 
(24) 



#o = 100 sech 1.5-500 J tanh 1.5 
= 42.510 - 452.58 h. 

Pq = EqIq. 

I = h sech vl + Ey tanh vl, 

= 0.42510 7o + 0.181030. 

E' = 100 sech 1.50 = 42.510. 

E — Eo Iq 



Reg. = 



En 



0.200 cosech 1.5 



0.0939284 - 7 
P= 100 7. 



Efficiency = -=£■ 



P' = line loss = P - P . 



TABLE I 
Calculation of Performance of Leaky Line 



h 


0.0 


0.01 


0.02 


0.03 


0.04 


0.05 


0.06 


0.07 


0.08 


0.09 


452.58 /„ 


0.0 


4.526 


9.052 


13.577 


18.103 


22.629 


27.155 


31.681 


36.206 


40.732 


E a 


42.510 


37.984 


33.458 


28.933 


24.407 


19.881 


15.355 


10.829 


6.304 


1.778 


P» 


0.0 


0.37984 


0.66916 


0.86799 


0.97628 


0.99405 


0.92130 


0.75803 


0.50432 


0.16002 


0.4251 /, 


0.0 


0.00425 


0.00850 


0.01275 


0.01700 


0.02126 


0.02551 


0.02976 


0.03401 


0.03826 


/ 


0.18103 


0.18528 


0.18953 


0.19378 


0.19803 


0.20228 


0.20654 


0.21079 


0.21504 


0.21929 


Reg. 


0.0 


0.11915 


0.27053 


0.46928 


0.74172 


1.13822 


1.76843 


2.92539 


5.74366 


22.9101 


P 


18.1030 


18.5283 


18.9532 


19.3783 


19.8034 


20.2285 


20.6536 


21.0787 


21.5038 


21.9289 


Eff. 


0.0 


0.02050 


0.03531 


0.04479 


0.04930 


0.04914 


0.04461 


0.03596 


0.02345 


0.00730 


P' 


18.1030 


18.1485 


18.2840 


18.5103 


18.8271 


19.2345 


19.7323 


20.3207 


20.9995 


21.7689 



The data obtained in the above table of calculations are shown 
plotted in Figs. 3 and 4. 



NUMERICAL ILLUSTRATION 



19 









































Amp 


Volt 








































0.215 




















































































0.210 


48 




























S\ 






















































0.205 


40 
























































'Po 


























0.200 


32 










































































\ P ° 








0.100 


24 
































































E> 


















0.190 


16 


















































































0.185 


8 


















































































0.180 






1.0 



0.8 



O.fi 



0.4 



0.2 



0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 

Load Current I o , in Amperes 
Fig. 3. — Characteristics of Leaky Direct Current Transmission Line. 
Generator Voltage Constant, 100 Volts. 



Eff. 




0.01 0.Q2 0.03 0.04 0.05 0.06 0.07 0.08 0.09 

Fig. 4. — Characteristics of Leaky Direct Current Transmission Line. 
Generator Voltage Constant, 100 Volts. 



20 



DIRECT-CURRENT LINE WITH LEAKAGE 



By equation (30) the maximum P occurs for 7 = 0.0469642 
amperes, which may be noted on the plotted curve and is, by (31), 
0.99821 watts. 

One more illustrative example will be considered. 

Take a line with constants, r = 5.0 ohms, and g = 2.0 X 10~ 5 
mhos, per mile. Then 

v = 0.0100, z = 500.0, and y = 0.00200. 

As load values, let E = 10,000 volts and J = 10.0 amperes. 
For this load, the following table gives values of generator voltage, 
current, power, line loss, and efficiency of transmission for various 
line lengths. 



TABLE II 



l 


E 


/ 


P 


P' 


Eff. 


miles 


volte 


amperes 


K.W. 


K.W. 





10,000.0 


10.0000 


100.00 





1.0000 


20 


11,207.4 


14.2275 


159.45 


59.45 


0.6272 


40 


12,864.4 


19.0257 


244.75 


144.75 


0.4086 


60 


15,038.0 


24.5877 


369.75 


269.75 


0.2705 


80 


17,814.9 


31.1365 


554.69 


454.69 


0.1803 


100 


21,306.8 


38.9348 


829.58 


729.58 


0.1205 



For line lengths greater than 40 or 50 miles, the generator 
voltage and current, and consequently power also, are seen to 
increase very rapidly. The example considered is, however, an 
exaggerated case inasmuch as the leakage coefficient, g, is several 
times larger than would be encountered in an actual system which 
was designed to transmit power at the voltage considered. The 
fine constants were arbitrarily selected so as to furnish results 
illustrating the properties of the solutions. 

In both this and the former illustrative example, the fine losses 
were of sufficient size in comparison with the total power to be 
obtained readily by subtracting the power at the load from that 
at the generator, thus furnishing no necessity for calculating the 
loss directly by equation (27). The above results are shown in 
Fig. 5.^ 

15. Effective Resistance of Line and Determination of Con- 
stants by Measurement. — For a direct current transmission 
system with no leakage the effective resistance of the line and 



EFFECTIVE RESISTANCE OF LINE 



21 



load at the generator end, as determined by the ratio E/I, is 
exactly equal to the load resistance plus the line resistance. For 
the leaky line such is not the case, the effective resistance depend- 





























Kilo 
Wat 


Kilo 
Volt 


.... 
Amp 


, 




























950 


19 
































900 


18 


30 
































































800 


1G 


32 


















































E/ 














700 


14 


28 
































































COO 


12 


24 
















1/ 




















Kff 




























500 


10 


20 


100 






























































■100 


8 


16 


80 
















































P/ 














300 


6 


12 


GO 


















































'?' 












200 


4 


8 


40 




















































Efl 










100 


2 


4 


20 











































































1(3 



32 48 64 i 

Line Length in Miles 



06 



Fig. 5. — Characteristics of Leaky Direct Current Transmission Line. Load 
at Receiver Constant; E = 10,000 volts and I = 10.0 amperes. 



ing upon the load resistance and the two line constants r and g, as 
well as upon the line length. 

The ratio of generator voltage to generator current defines the 
effective resistance. Utilizing equations (18a), after canceling the 
common factor 7 in numerator and denominator, one has 

n _ E _ Rp cosh vl + z sinh vl . _. 

I cosh vl + R y sinh vl 
= effective resistance from generator end as would 
be indicated by a Wheatstone bridge. 



22 DIRECT-CURRENT LINE WITH LEAKAGE 

The above equation, though complete, is scarcely as important 
as two special cases derivable from it. They are for the effective 
resistance when the load end of the line is short-circuited, and 
when the load end is open-circuited. The first may be derived 
directly from (37) by placing R equal to zero, but the placing of 
Ro equal to infinity (rather, allowing R to increase indefinitely) 
for the second case would lead to an indeterminate form. Dividing 
both numerator and denominator of (37) by Ro, or utilizing equa- 
tions (18b) gives 

P _ E _ cosh vl 4- Gpz sinh vl .__. 

/ Go cosh vl -f- y sinh vl ' 

from which, by placing G = 0, the desired result is obtained. 
Thus, from (37) for R Q = 0, 

R' = z tanh vl = short-circuit resistance ; (39) 

and from (38) for G = 0, 

R" = z coth vl = open-circuit resistance. (40) 

For infinite values of line length, the two expressions (39) and 
(40) as well as (38) approach the value of z, since the hyperbolic 
tangent and cotangent both approach unity for continuously 
increasing values of argument. Thus, for infinite line length, the 
terminal or load resistance has no influence upon the effective 
generator-end resistance of the system, which is then equal to 

z = y - = infinite line resistance. (41) 

Equations (39) and (40) are of importance in determining the 
values of r and g from actual measurement of line resistance. 
There are two quantities to be determined, and therefore two 
measurements taken under different conditions will be required. 
The two simplest conditions to utilize are obviously those men- 
tioned above, viz., short-circuited and open-circuited, and from 
these two resistance measurements, taken either by means of a 
bridge or by the voltmeter-ammeter method, numerical values of 
R' and R" will be furnished. 

From these values of R' and R" the values of r and g are to be 
determined. 



EFFECTIVE RESISTANCE OF LINE 
Multiplying (39) and (40), 

r'r" = 2 2 = r, 

9 
and by division of these same equations, 

^77 = tanh 2 vl; \ ~5Tt = tanh I Vrg. 

Thus, knowing the line length, I, 

z = J- = VWW', 

» n 



23 



v = Vrg = j tanh" 1 \-^r, 



r = vz = - l VrW tanh- 1 



g 



z l VR'R" 



tanh 



\ R" 



(42) 



(43) 



Equations (42) and (43) suffice to accurately and completely 
determine the fundamental line constants from the two measure- 
ments, R f and R", made under the specified conditions. The 
above method, or its equivalent, must be used in order to deter- 
mine the two fundamental line constants accurately when the 
total line resistance and leakage are relatively large, or to be more 
exact, when the product of these totals exceeds a certain definite 
amount depending upon the accuracy desired. 

The ordinary approximation methods for determining the line 
constants consist in taking 



'■-i*> 



9 = 



Rt = rl — R', 
1 



IR" 1 



G t =gl = 



R'" 



(44) 



which are based on the assumptions that when the line is short- 
circuited for measuring R' all the current traverses the entire 
line — that is, is of uniform value along the line length; and that 
when the line is open-circuited for measuring R", the potential 
difference between conductors is the same throughout the length. 



24 DIRECT-CURRENT LINE WITH LEAKAGE 

The approximation (44) is equivalent to assuming that in (43) 

tanh -1 y-p77 ma Y De replaced by y p77> 

which will obviously introduce considerable error for values of 
the radical greater than two-tenths. It is also apparent, by (42), 
that the value of the ratio 

K 
R" 



V R" 



is dependent only upon the product vl = y/R t Gt. 

The following table will show the magnitude of the error which 
the approximation method would introduce under various con- 
ditions: 

TABLE III 
The Inverse Hyperbolic Tangent 



T R" 


tanh"« y ~ = ^Rfit = vl 


|/1 

▼ R" 


tanh-i y j^i = ^Rfit = rf 


0.00 


0.00000 


0.55 


0.61839 


0.05 


0.05004 


0.60 


0.69314 


0.10 


0.10033 


0.65 


0.77530 


0.15 


0.15114 


0.70 


0.86729 


0.20 


0.20273 


0.75 


0.97295 


0.25 


0.25541 


0.80 


1.09861 


0.30 


0.30952 


0.85 


1.25615 


0.35 


0.36544 


0.90 


1.47221 


0.40 


0.42365 


0.95 


1.83180 


0.45 


0.48470 


1.00 


Infinity 


0.50 


0.54931 











By the use of the above table the error introduced into the 
determination of the fundamental line constants by the use of the 
approximation method may be determined as soon as the measure- 
ments on which the computations are to be based are available. 

16. Combinations of Leaky Lines. — In all of the preceding 
the discussion is applied to a line supplied with power at one end 
and delivering energy at the other end — a condition which per- 
mits the enumeration of practically all of the equations covering 
the phenomena of transmission of energy over leaky lines. The 
conditions named in the heading of this paragraph do not permit 



SINGLE GENERATOR SUPPLYING TWO SEPARATE LOADS 25 

of such ready and general treatment, though the general method 
of attack may be indicated. Assuming, of course, that each 
individual section of line in such a system is uniform throughout 
its length, the fundamental differential equations (3) and there- 
fore the general solutions (14) still apply if the integration con- 
stants E and 7 are correctly interpreted. 

17. Single Generator Supplying Two Separate Loads. — As a 
first special case, consider a generator of constant voltage E 
supplying power to two separate loads at A and B, the connecting 
transmission systems being leaky. Figure 6 serves as illustration. 
The two sections of line will be defined by the quantities, 

Section 1, r h g h l h or z h v 1} l h y x = 1/fc. 
Section 2, r 2 , g 2 , k, or z 2 , y 2 , h, y% = 1/%. 

The loads at A and B may, however, be specified in different ways. 
Suppose, for instance, that they be specified by means of equiva- 



£ 



Section 1. I I" Section 2. 

Fig. 6. — Single Generator Supplying Two Separate Loads. 

lent resistances, R a and Rb, respectively. To determine all currents 
and voltages in the system when the generator voltage is equal to 
E, proceed as follows: 

The equivalent resistance R" of the load Rb and line section 2, 
as measured at A, would be given by (37) as 

„„ _ Ea _ Rb cosh V2I2 + 32 sinh v 2 k ,.-. 

I" cosh vtk -+- RbV2 sinh vJq 

The above defined equivalent resistance is in parallel with the 
load resistance R a , and thus the equivalent resistance of the two 
combined is 



p, _ Ea _ RgR 

a ~ V - Ra + R'" 

G' = 1/R'. 
Then immediately, by (19), 



(46) 



/' = (47) 

R' cosh vih + 0i sinh V\li • 



26 



DIRECT-CURRENT LINE WITH LEAKAGE 



Also, 



Ea = R I , 




j E a 
la — q-J 
Ita 




Jfl ■"<• Jl J 




Again, by equation (19), 

J E a 


lib cosh ^ + z% sinh V2I2 ' 


Eb = Rblb- 





(48) 



(49) 



Finally, at the generator, by equation (12), 

I = T cosh vik + E a yi sinh vik. (50) 

Equations (45) to (50), which apply to the particular case under 
discussion, are arranged in convenient form for computation. 
They are not entirely independent inasmuch as the results obtained 
from one equation are introduced into one following. 

It is of interest to note the quantities which may be specified at 
will in such a transmission system. Assuming the line constants 
as fixed and the loading points A and B established, we may 
specify, for example, 

(1) Eb, h, and I a , 

(2) E b , E a , and I a , 

(3) R a , Rb, and E, 

but not such combinations as 

(4) E a , E b , and h, 

(5) E a , h, and I a , 

for in the latter it is readily seen that the quantities are not in- 
dependent. 

It is impossible to treat here an extended number of special 
cases, but the example just given may serve as a help in deciding 
upon a method of attack. 

18. Single Load Supplied from Two Power Sources. — A 
different form of problem is that in which a single load is supplied 
from two sources of power over lines possessing leakage, as shown 
in Fig. 7. 

Again, the problem may be stated in several different ways, 
some of which lead to impossible solutions. For example, with 



SINGLE LOAD SUPPLIED FROM TWO POWER SOURCES 27 

fixed generator voltage and a short-circuit at the point of loading, 
the current I would be the sum of the two currents supplied by 
the two separate short-circuited lines, which, by equation (19) 
when R = 0, would be 

E^i cosech vj,i + E 2 y 2 cosech v 2 k. 

It would be absurd, with fixed generator voltages, to specify a 
load current 7 greater than the above which could be supplied 

-e i' i" ^ i 9 





Section 1. Section 2. 

Fig. 7. — Single Load Supplied by Two Generators. 

over a short-circuit. The same is true in the discussion of a 
similar system without leakage. 

As a concrete case, let the resistance of the load, R = -p, be 

jo 

specified. All electrical quantities for given values of E\ and E 2 

are to be determined. Let the convention as to positive direction 

in each circuit be as indicated by the arrows. Then, with the 

notation shown in the diagram, 

i—r+r; ^ 1 (51) 

Eo = RoI r + RqL . J 

The two currents Y and I" will be taken as the two unknown 
quantities whose determination constitutes the main feature of the 
solution of this problem. By equation (14), 

Ei = E cosh Vik -f- I'zi sinh vh, 
E 2 = ^o cosh V2I2 + 1'% sinh v 2 h- 

Substituting for E its value as given by (51) and collecting terms, 
(#0 cosh Vik + Zi sinh vji) I' + (R cosh vik) I" = E h 
(Ro cosh v 2 k) V + (Ro cosh v^k + Z2 sinh v 2 l 2 ) I" = E 2 . 



(52) 



In (52) there are two simultaneous equations in the two unknown 
quantities I' and I", which suffices for their determination. The 
solution by determinants yields 



28 DIRECT-CURRENT LINE WITH LEAKAGE 

,,_ (Ro cosh ttk-l-frBmh V2U) Ex — (Ro cosh vM Ei 

(Ro cosh vrfi + z x sinh v x l\) (Ro cosh v t l% -+- z% sinh fj/2) — #o 2 cosh v x l x cosh y^ ' 

,„_ (Ro cosh r^! +z t sinh vJi) E 2 — (Ro cosh ink) Ei 

(Ro cosh vJi +zi sinh i'i£i) (Ro cosh 1 j/ 2 +2jsinh »; 2 ^a) — .fto'cosh 1^ cosh ^/j ' 

. (z% sinh t>th) Ei -\- (z x sinh v\k) E t 

(Ro cosh fill + Z\ sinh v x U ) (.#0 cosh t^ + zj sinh t^ 2 ) — .fto'.cosh Vik cosh i^a 



(53) 



The voltage at the load is R I , which becomes, after expanding 

the denominator of the last equation of (53) and multiplying by R , 

En = RoIq == 

(22 sinh v 2 k) Ei + (z x sinh v x l x ) E 2 ._.. 

iZ (54) 

22 cosh v\l\ sinh t^ + Zi cosh v 2 k sinh »iZi + -5- sinh fliZi sinh fl 2 k 

Ho 

19. Unloaded Line with Double Power Supply. — Having 
determined the components of the load current by (53) and the 
load voltage, the determination of all other quantities is accom- 
plished with facility by the application of the several general 
equations applying to simple circuits. The results in equations 
(53) and (54) are perfectly general, holding for all values of load 
resistance, and they therefore lead directly to a method of deter- 
mining the line current and voltage at any point when no load is 
present by the simple device of letting the resistance R a approach 
infinity in value. 

Expanding the denominator in (53), dividing both numerator 
and denominator by Ro, and discarding all terms which reduce to 
zero on account of the infinite value of R , the resulting equations 
for the line currents become 



Ei cosh v 2 U — Ei cosh WiZi 



r/1 . 

_Uo = 00 Z2 cosh V\l\ sinh Vzk + 2 X cosh i^k sinh Vih ' 
Ez cosh Vik — Ei cosh vjv 



(55) 



_Uo= 00 Z2 cosh V1I1 sinh v?k + Zi cosh v?k sinh V1I1 ' 
while from (54), the voltage at the point in question is immediately 

E "I m E1Z2 sinh V2I2 + E 2 zi sinh vji . . 

°_Uo= °° 22 cosh Vik sinh v 2 k + 21 cosh v 2 h sinh vj,i 

The results expressed by equations (53) to (56) are of interest 
and importance. In the case of the loaded line as covered by 
equations (53) and (54), the distribution of current and voltage 
along either portion of the transmission system is conveniently 



NUMERICAL ILLUSTRATION 29 

obtained by use of equation (14), counting distance from the point 
of loading and using the results obtained from (53) and (54) as 
initial quantities. Since the solutions as given are general, either 
positive or negative numerical values may be assigned to the 
electromotive forces E x and E 2 . The meaning of the algebraic 
sign in a numerical solution resulting from the use of these equa- 
tions follows the convention usual in any application of Kirchhoff 's 
Laws where a direction as indicated by an arrow is arbitrarily taken 
as positive. 

Equations (55) and (56) applied to a uniform, unloaded line 
serve immediately for the determination of the distribution of 
voltage and current. In a non-uniform or composite line con- 
taining two sections of line with different constants, the solution 
must be made first for the voltage and current at the junction 
point, and then, utilizing equation (14) by the method mentioned 
in the preceding paragraph, for any desired point throughout the 
length of either section. Of course, in the case of the uniform, 
unloaded line whose constants are the same throughout the entire 
length between the two sources of power supply, the lengths k and 
h, must be so taken that their sum is equal to the total line length. 

Naturally, if the line is uniform throughout, a simplified form 
of expression will result from either (53) and (54) or from (55) and 
(56), for in this case Zi = z 2 , and the denominators in (55) and (56), 
for example, reduce to a single function. For a uniform line whose 
total length is h -J- k, with constants v and z per unit length, 



r /l _ Ei cosh vk — Ei cosh vk 

Jfl 0=O o _ z sinh v (k + k) 

E 2 cosh vk — Ei cosh vk 

z sinh v (li + k) 

Ei sinh vk + E 2 sinh vli 

sinh v (l x -f- k) 



"'1 " 



(57) 



*1^-" 



The denominator is constant for all points selected, since h + k is 
equal to the total line length, and therefore constant. 

20. Numerical Illustration. — A numerical illustration will be 
given here of the distribution of current and potential difference 
along a uniform, unloaded line, utilizing equations (57) and (58). 

Let the total line length be 100 miles and the line constants be 

r = 10.0, g = 10~ 5 , per mile. 



30 



DIRECT-CURRENT LINE WITH LEAKAGE 



Then 

Z = Zi = «2 = 10 3 . y = Vl = y 2 = 10- 3 . 
v = Vl = 2>2 = 0.0100. 

The denominator of (57) is 10 3 sinh 1.00 = 1175.20. 
The denominator of (58) is sinh 1.00 = 1.17520. 

Let Ei = 100 volts and E 2 = 80 volts, both positive, and 
therefore acting in the direction indicated by the arrows in Fig. 7. 
The following table contains the calculations for determining the 
distribution of line voltage and current. 



TABLE IV 

Line with Double Source of Power Supply 



h 


00 


10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


h 


100 


90 


80 


70 


60 


50 


40 


30 


20 


10 


00 


*l 


0.00 


0.10 


0.20 


0.30 


0.40 


0.50 


0.60 


0.70 


80 


0.90 


1.00 


•1. 


1.00 


0.90 


0.80 


0.70 


0.60 


0.50 


0.40 


0.30 


0.20 


0.10 


0.00 


Ej sinh vli 


0.00 


8.01 


16.11 


24.36 


32.86 


41.69 


50.93 


60.69 


71.05 


82.12 


94.02 


E\ sinh Wj 


117.52 


102.65 


88.81 


75.86 


63.66 


52.11 


41.08 


30.45 


20.13 


10.02 


00.00 


Sum 


117.52 


110.66 


104.92 


100.22 


96.52 


93.80 


92.01 


91.14 


91.18 


92.14 


94.02 


Eo 


100.00 


94.16 


89.28 


85.28 


82.13 


79.81 


78.29 


77.55 


77.59 


78.40 


80.00 


Ei cosh vlt 


154.31 


143.31 


133.74 


125.52 


118.55 


112.76 


108.11 


104.53 


102.01 


100.50 


100.00 


E t cosh tli 


80.00 


80.40 


81.61 


83.63 


86.49 


90.21 


94.84 


100.41 


106.99 


114.65 


123.45 


Difference 


74.31 


62.91 


52.13 


41.89 


32.06 


22.55 


13.27 


4.12 


-4.98 


-14.15 


-23.45 


10»X/' 


63.23 


53.53 


44.36 


35.64 


27.28 


19.19 


11.29 


3.51 


-4.24 


-12.04 


-19.95 



The distribution of current and voltage along the line as deter- 
mined in the above table of computations is shown graphically in 
Fig. 8. Naturally, with an unloaded line supplied with energy 
at both ends in the manner considered in this illustration, the 
direction of current flow reverses at a point coinciding with the 
minimum of line voltage. 

21. General Network of Transmission Systems with Loads 
at Various Points. — Of such a system as mentioned very little 
can be said unless a large number of special cases be taken under 
consideration, and as even then the field of possibilities could in 
no way be covered, the solution of such problems, if an exact 
solution be desired, must be left to the ingenuity and mathematical 
ability of the individual. In general, unless the problem be stated 
in a manner at variance with the physical possibilities (for instance 
by specifying a load current at some point greater than could be 
supplied by generators of the selected voltage over the trans- 
mission systems involved), it will always be possible to obtain a 



GENERAL NETWORK OF TRANSMISSION SYSTEMS 31 

solution. Kirchhoff's Laws will in general supply a number of 
current equations. If the generator electromotive forces are 
specified and the loads represented by their equivalent resistances, 
the scheme adopted in arriving at equations (51) to (54) will serve 
in a majority of cases. The known electromotive forces are 











































Eo 
100 










































Volt 
98 










































96 










































94 










































92 










































90 










































88 










































86 










































84 


























I' 
















82 


















E> 
























80 










































78 










































76 










































74 










































72 













































10 



Fig. 8. 



20 



80 



90 



100 



I' 

milU- 
amps. 



30 40 50 60 70 

Distance, I z , from 100 Volt Generator. 

Distribution of Voltage and Current along an Unloaded Leaky 
Line when Supplied with Power at both Ends. 
E x = 100 and E 2 = 80 volts. 



expressed in terms of the load resistances and the unknown cur- 
rents. A solution of the simultaneous equations so obtained 
expresses the unknown currents in terms of the known generator 
voltages and the load resistances. When applied to a single load 
fed by a number of transmission systems, this procedure is very 
simple — leading to as many simultaneous equations as there are 
transmission systems involved. Figure 7 illustrates a case having 
two such supply systems. With the possibility of using negative 
values for the generator voltage's in the general solution, the 
generators may virtually be considered as motors (or loads) of the 
voltage selected, and the range of applicability of the general 
solution thereby extended. 



32 DIRECT-CURRENT LINE WITH LEAKAGE 

Even with three systems of transmission involved in a network, 
as above mentioned, the expressions both before and after the 
solution of the simultaneous equations are quite complicated, and 
with an increase in the number of systems involved or with an 
addition of intermediate loads the complexity of the mathematical 
development increases enormously. 



CHAPTER III 

PERIODIC AND ALTERNATING QUANTITIES. FOURIER'S 
SERIES AND ANALYSIS OF PERIODIC CURVES 

22. General. — The subject of this chapter is one which has 
been treated very many times. It is discussed here in order that 
the present volume may be rendered complete and may present to 
the person engaged in any investigation the means for obtaining 
with ease and dispatch the analysis of any given curve which is 
capable of being represented by a Fourier's Series. The matter of 
analysis by means of a finite number of ordinates, which heretofore 
has been considerably neglected, is also discussed. As a matter 
of interest a form of mechanical analyzer which was designed by 
the writer to facilitate the exposition of the theory of analysis is 
described. 

23. Periodic Quantities and Fourier's Theorem. — A quantity 
periodic in time may be described as one which passes recurrently, 
in equal intervals of time, through the same series of values. 

An alternating quantity is a special form of periodic quantity, 
in that in successive equal intervals of time it passes through a 
series of values which are both positive and negative. 

A special form of Fourier's Theorem, which is sufficient for the 
purpose in hand, states that any continuous, single-valued, finite, 
and periodic function may be represented by a series of trigono- 
metric sines and cosines — the terms in such a series having fre- 
quencies which are multiples of the fundamental frequency of the 
periodic function. 

24. Use of Fourier's Series. — The utility of Fourier's Series 
in the treatment of alternating current problems is based on the 
fact that by the separation of a periodic quantity into its constit- 
uent terms, a number of simple harmonically varying quantities 
are obtained which, in further calculation, may each be treated 
separately by means rigorously applicable only to such simple 
harmonically varying quantities. Methods of employment of the 
results of such an analysis will be treated in a later chapter; 

33 



34 FOURIER'S SERIES 

this chapter deals only with the methods for separating a given 
periodic function into the constituent terms. 

Usually the given periodic quantity will be specified graphically 
by a curve, or by a set of numerical data from which such a curve 
may be plotted. Thus, y may be considered as a function whose 
value is known for any value of 0. 

25. Determination of the Unknown Coefficients by Integra- 
tion. — Let the scale of the independent variable be so selected 
that the length of the period is equal to 2 ir, and for the independent 
variable to this scale use the quantity or angle 0. Then we assume 
that 

y = f (0) = A + Ai cos + A 2 cos 2 + ■ ■ • + A n cos n0 

+ • • •+fiisin0 + £ 2 sin20 + - ■ ■ + B n sin ri0 + • • • (1)' 

the problem then being to determine the values of the unknown 
coefficients, A , A h A 2 , etc.; B h B 2 , B 3 , etc. 

In the usual manner, the details of which need not be discussed 
here, we find that by multiplying both sides of equation (1) by 
cos nd dd and integrating between the limits and 2 t, 



-If 

7T Jo 



2x 

y cos nd dd, (2) 



except for the special case of n = 0, when the expression 

1 r 2r 

A = ± ydd (3) 

must be used. 

By multiplying throughout by sin nd dd and integrating, 

B n =-f 'yswnddB. (4) 

IT Jo 

From equation (2) it is seen that the coefficient A„ is equal to - 
times the area under the auxiliary curve 

y' = y cos n0, (n ^ 0), 



when measured between the limits and 2ir. Similarly, B n is 

er the auxiliai 
y" = y sin nd. 



equal to - times area under the auxiliary curve 



THE COSINE TERMS 35 

26. Use of the Auxiliary Curves, y' = y cos n6 and y" = y sin 
n8. — If, from the original curve, these auxiliary curves are 
calculated and plotted, the areas enclosed could be measured with 
a planimeter and the values of the various coefficients thereby 
determined. If a number of equidistant ordinates are known for 
the original curve, the area of the auxiliary curves may be deter- 
mined approximately by taking the average height of ordinate of 
the auxiliary curve as computed (y' or y") and multiplying by the 
length of the base, 2 t. Dividing the area so obtained by r as in 
equations (2) and (4) and by 2 w as in equation (3), we have the 
simple result 

A = (average y), 

A n = 2 (avg. y cos nd), (n 9^ 0), (5) 

B n = 2 (avg. y sin nd), 

subject to the errors arising from the approximate manner of 
obtaining the numerical value of the integral by the use of the 
average value of a finite number of ordinates. 

27. Determination of Coefficients from a Finite Number of 
Ordinates. — The determination of the various unknown coeffi- 
cients from a finite number of ordinates will now be considered. 
The given curve is to be represented by a series of the form of 
equation (1) in which the number of trigonometric terms may be 
unlimited. 

Divide the wave length, or length of curve which is to be rep- 
resented by Fourier's Series, into t equal parts, numbering the 
ordinates 0, 1, 2, 3, ... (t — 1), t. For the fundamental fre- 
quency, the total wave length is 2 tt radians, and therefore ordinate 
number m has an argument or angle of 

2 rax 



t 



radians. 



y m =A Q -\-A 1 cos— — hA 2 cos2 — — (_ . . . _j_^ nC0Sn __-j_ . . . 

. D . 2wtt D . n 2rmr D . 2 rax , , A * 

-f-BiSin— — h-D 2 sin2 — - — \- • • • +B n smn— - — [- • • • . (6) 

I L L 



28. The Cosine Terms. — If we multiply equation (6) through- 
out by cos p — — , where p is any positive integer, and take the 
summation of all ordinates over a complete wave length, 



36 FOURIER'S SERIES 

2rox A "g 2wt , A m 4l 2mir 2rmr 

cos — : — 



2, j/mcosp— — = Ao v cos P —, — VM 2j cos P ~i 

m-l l m-l l « 

. . . <^\ 2wi7r 2mir . 

+ • • • + A n 2, cosp— — cosn — f- 

m — 1 

+ #i V cos p — 

m-l f 



2 mi . 2 rax 
sin 



ro-< 



2wir . 2W7T 



+ • • • + B n 2j cos p — — sin n — — + • • • . (7) 

m — 1 

The summation constituting the coefficient of a general B, or 
sine, term may be represented by 

S2 rnir . 2 mir 
cos p — — sin n — t— > 

m = l ■ * 

which by the expansion 

cos a sin /3 = | [sin (a + /3) — sin (a — /S)] 
reduces to 

„„ = I "| [sin (? + f 2 "" r - sin (P^L2mj-j (g) 

The above summation vanishes for all integral values of p, n, 
and £, as can readily be seen from the fact that the summation of 
sines is equivalent to the summation of the projections on the 
vertical axis of t equal length vectors in a plane spaced under equal 
angles. Therefore in equation (7) there remain only the cosine, 
or A, terms. 

The summation constituting a general cosine coefficient is 



Zn = \% COS p 



2 m-w 2 mir 

cosn 



t t 



which by the expansion 

cos a cos jS = \ [cos (a + /3) + cos (a — $)\ 



reduces to 



1 m=l r 

-|X,[« 



. (P+")* m * + cos (p-»)2"« 1 . (9) 



In (9) the summation of each term taken separately will vanish 
unless 

p + n = it, or p — n — it, 



THE SINE TERMS 37 

that is, unless 

n = it — p, or n = p — it, 

in which i represents any integral number, i must assume all 
values both positive and negative in order that every positive 
value of n be represented. 
Then, since 

2, cos — — = t, (10) 

equation (7) reduces finally to 

m = t cy j. i = °° r- -j 

5) VmCOSp -^ = 2 X A ^~P> + A (P~iO \j (11) 

m=l t=— oo L J 

in which any A with a negative subscript is meaningless and 
therefore to be discarded. 

29. The Sine Terms. — Proceeding in a similar manner for 
the establishment of expressions containing only the B coefficients 

we multiply equation (6) throughout by sin p — — and take the 

t 

summation over a wave length. 

V-* . 2 rax A v^ . 2 rax . . -^ . 2rmr 2 rax 
£ y m sin p —r- = A Q £ sin p — \- A x £ sin p —r- cos — — 

m = l m = l m = l 

. . v^ . 2 rax 2 rax . 

+ • • • + A n £ sin p — — cos n — — h • • • 

TO=1 
, „ V* • 2W7T . 2W7T 

+ £i 2, sin p — — sin — — 

ra = l 



rra=< 



2 rax . 2 rax 



+ • • • + B n 2j snip — — sinn— — + • • • . (12) 



771 = 1 



In a manner identical with that used for equation (8) it may be 
shown that the summations constituting the coefficients of the 
various cosine, or A, terms will all vanish separately when n and 
p have integral values. 

The coefficient of a general sine term is 



m=l 



2 2 rax . 2 rax 

smp —j— sinn — — i 



m=l 



38 FOURIER'S SERIES 

which by the expansion 

sin a sin = \ [cos (a — /3) — cos (a -f /3)] 

reduces to 

1 V T (p — n) 2 mir (p + n) 2 mxl „ ON 
»» = 2 i cos j cos j ' ( 13 ) 

In equation (13) the separate summations entering v n are always 
zero unless 

p — n = it, or p + n = it; 
that is, unless 

n = p — it, or n — it — p, 

where, as before, i must take all negative and positive values in 
order that all positive values of n be represented. 

i= • • • , -3, -2, - 1, 0, +1, +2, +3, • • • 
Thus, by equation (10), 

2 ?/ m sin p -^ = x J) B( P - it ) - B (it -p) , (14) 

in which, as before, negative subscripts are to be neglected. 

30. Interpretation of Results — Cosine Terms. — Equation 
(11) may be written 

{=30 p ~1 O m= . t O 

X I ^w-p) + ^(P-fl)J = f%ymCOsp -J— (15) 

Obviously, the right hand member of equation (15) is twice 
the average value of t ordinates, equally spaced over a wave length, 
of the auxiliary curve 

y' = y cos pB, 

which is exactly the same as the right hand member of the second 
of equations (5). Consequently, it is apparent that by the use of 
the finite number of ordinates, t, the numerical value of twice the 
average ordinate of the auxiliary curve is not equal to the unknown 
coefficient A p alone, but to the summation of all the A coefficients 
of the form indicated by the left hand member of (15) which may 
be present in the original curve whose analysis is sought. In case 
an infinite number of ordinates are used, the process of averaging 
becomes equivalent to an integration, and thus equations (5) and 
(15) are equivalent for t = infinity. 



PARTICULAR ORDERS OF HARMONICS 39 

To be specific, suppose for some particular curve we have 
measured 18 ordinates over a complete wave length. Also sup- 
pose that for the purpose of determining the cosine term of the 
fifth harmonic we have formed the numerical value of the right 
hand member of equation (15). Then, t = 18 and p = 5, so that, 
neglecting negative subscripts, we have 

A5+Ai3+A23+A31+j4 4 1+449+Ab9+ • * ' = jg 5/ Vm C ° S 5 "Ts"' 



Using the expression (5) for this particular case would thus give 
a correct value for A 6 only in the event of all the A coefficients of 
the orders A13, A23, A31, A41, etc., being equal to zero, that is, being 
absent in the particular wave shape under investigation. 

31. Interpretation of Results — Sine Terms. — Equation 
(14) for the sine terms may be similarly treated. 

«=? r l rr* T 2 ^rt • 2m7r 

X B (p . it) - B lU - P ) \= 7 X Vm Sm V ~T' ( 16 > 

t'= — 00 L -1 m = l 

which may be compared with the third of equations (5). 

Applying (16) to the numerical example in the preceding para- 
graph, we note that 

B 6 - B13 + £23 - £31 + Bn - + • • • = — V y m sin 5 -==-• 

Similar conditions for the exact determination of B& are neces- 
sary here as were required for the exact determination of A 5 . 

32. Particular Orders of Harmonics. — When p is zero or 

some multiple of -x- 

For the purpose of determining A we let p = 0, and then the 
multiplier cos p — - — in (15) is always unity. The result is then 

V 

1 m=t 



or 



A Q -Mo + At 4- A t + An + A 2t + • • • =7 X y*» 

1 «-i 

1 m= t 

A + A t + A 2t + A zt + • • • = 7 X Vm ' ( 17 ) 



w=i 



which coincides with the first of (5) except for the disturbing 
terms, A t , A^, A 3t , etc. 



If p is an even multiple of ~ it is also a multiple of t, the case 



40 FOURIER'S SERIES 

If p is some multiple of t, the multiplier cos p — - — again 

I 

reduces to unity and the result of the summation is the same as 
in (17). 

In regard to the sine terms, if p is zero or any multiple of t, it is 
obvious that the right hand member of equation (16) disappears. 
Building up the right hand member for p equal to t or any multiple 
of t, by giving negative and positive values to i, we obtain 

B -B +B t -B t +B 2t -B 2l + =- £ i Jm smkt^=0, (18) 

m — 1 

where k = any integer. 

If p is an e 
just discussed. 

For p an odd multiple of -x, equation (15) reduces to 

A t + Am + Am + • • • = j^,y m cos p — - — (19) 

2 2 2 l w = l l 

This equation reduces to 

A t + Au + A 5J + - • ' = -:X{-\) m y m (19a) 

2 2 2 l m = l 

which indicates that the determination of any A coefficient whose 

subscript is an odd multiple of - is vitiated by the presence in the 

given curve of all other coefficients of the same type. 

Obviously, the right hand member of (16) is zero when p is an 

odd multiple of ~. Expanding the left hand member, we have, 

m 

when p = any odd multiple of x, 

B»-B t + Bu-Bu+ - • • • =j%y m smp^=0. (20) 

2 2 2 2 l m = l l 

33. Limitations to the Use of a Finite Number of Ordinates. — 
The above development as summarized in equations (15) and (16) 
is very important in its relation to the actual carrying out of an 
analysis. It is apparent that if a given curve actually consists of 
an infinite number of terms, or harmonics, an analysis using only 



SUMMARY 41 

a finite number of ordinates will not give a correct result. If t 
ordinates are used, the lowest order harmonic which enters to 
vitiate the determination of the pth harmonic is that of order 
t — p, as long as p is less than t. Thus, if it is known beforehand 
that a certain curve contains harmonics up to and including a 
certain order, to accurately determine any or all of the cosine terms 
it will be necessary to use a number of ordinates at least twice the 
order of the highest harmonic known to be present. By equation 
(20) it is evident that to accurately determine all of the sine terms 
it will be necessary to use a number of ordinates at least one greater 
than twice that of the order of the highest harmonic known to be 
present. This arises from the fact that (20) fails to determine the 

B coefficient of order =• 

34. Summary. — Although, as was pointed out in the pre- 
ceding paragraph, it is impossible to secure a rigorously true 
analysis of a curve containing an infinite number of harmonics by 
the use of a finite number of ordinates, this fact need cause the 
engineer, interested primarily in results accurate only within the 
limits set by the possibilities of physical measurements, much 
concern. As a rule, in any curve representing physical phenomena 
the magnitude of the various harmonics present in a periodic curve 
decreases with increase in order, and if so, it will usually be possible 
to use a sufficient number of ordinates to cause equations (5) to be 
sensibly true. 

Though there is no intent of discussing here the question of 
convergence of Fourier's Series or the many very interesting 
properties of such, it may be well to remark that by the selection 
of a certain number of ordinates covering one wave length, a 
similar number of points are established on the original curve 
through which the curve represented by Fourier's Series resulting 
from the analysis will pass. This will be true when the subscripts 

to the A or B terms are taken to a number not to exceed ~ , when 

the ordinates are numbered according to the scheme outlined. 
Thus by selecting, say, 36 ordinates (t = 36), by equations (15) 
and (16) we may determine 

A , A h A 2 , . . . An, and Ai 8 , 
as well as 

Bi, Bi, . . . Z?ie, and Bn, 



42 FOURIER'S SERIES 

making in all 36 independent coefficients determined from the 
36 ordinates. As pointed out in section 32, equation (20), it is 
impossible to determine B^ in this case. The Fourier's Series 
thus established will pass exactly through the 36 points selected, 
so that the only possibility for difference between the original curve 
and that resulting from the analysis is in the shape of the arch 
between ordinates. Again, if 35 ordinates equally spaced over a 
wave length were used, values of 

A , Ai, A if . . . A i6, and An, 
B h B i} . . . Bi 6 , and B„ 

would be determined — this time 35 independent coefficients from 
the 35 ordinates — which would pass a curve exactly through the 
35 points. The two sets of coefficients would in general be differ- 
ent, depending upon the magnitudes of the harmonics higher in 
order than 17 present in the original curve. 

As far as engineering and physical sciences are concerned, 
analysis with a finite number of ordinates can be made sufficient, 
for in a majority of cases the original curve is specified only by a 
number of plotted points — the shape of the arch between points 
being unknown at best. In such cases an analysis as above indi- 
cated will give the simplest curve which will represent all of the 
data. Again, in the analysis of curves which are drawn continu- 
ously by some recording device, the accuracy of an analysis will in 
general be affected more by the inability to secure perfect measure- 
ment of instantaneous values than by the failure of the methods 
given to yield exact results; Illustrative examples of analyses will 
be shown later. 

35. Mechanical Analysis Based upon Integration Processes. 
— The planimeter is a device which gives directly the numerical 
value of an integral as typified by the area enclosed by a curve. 
What a mechanical analyzer must accomplish is to give, by means 
of a planimeter, the numerical values of integrals of the form indi- 
cated in equations (2), (3), and (4). This has been done in several 
ways, one of which is described here. The apparatus is illustrated 
schematically in Fig. 9. 

In the drawing, y = f (x) is the given curve to be analyzed, and 
to fix our ideas let us suppose that the coefficient B n is desired. T 
is a tracing point with which to follow the given curve. A is the 
main body carrying the tracing point, which is constrained to a 



MATHEMATICAL THEORY OF ANALYZER 



43 



parallel motion by the mechanism B. A rack, at the end of which 
is a slide cross-head, is free to move backwards and forwards 
through the fixed guide G. Thus, the movement of the rack is 
exactly equal to the x movement of the tracing point T. 




Place planimeter point in 
Intersection of_two right angle 
slots, P. 



Fig. 9. — An Harmonic Analyzer. 



The motion imparted by the rack to the gear D is transmitted 
through the gear-train M to the face-plate in which is a projecting 
pin at a distance R from the center as shown. Let the gear-train 
be so dimensioned that the radius R makes n revolutions to one 
movement of T from to 2 t (one wave length) on the x axis of 
the given curve. The action of this pin in the slotted cross-head 
gives a horizontal movement to the bar E dependent upon either 
the sine or cosine of n times the horizontal displacement of the 
tracer T. 

To the body A is attached the slotted arm H which moves in 
a vertical direction an amount equal to y on the given curve. 
Therefore a planimeter point placed in the right-angle intersection 
of the slots in H and E will have a motion made up of the two 
components as described. 

36. Mathematical Theory of Analyzer. — Take supplementary 
axes as shown, corresponding to the center of the circle described 



44 FOURIER'S SERIES 

by R and to the position occupied by the slot in A when the point 
T is on the x axis of the given curve. Let the radius R be in its 
extreme position to the left when T is at the origin of the given 
curve. 

Denote coordinates of the point P with reference to the supple- 
mentary axes by x' and y'. Then 

y' = y, 

x' = — R + {R — R cos 0) = —R cos <£, 

where </> is the angular displacement of the radius R in a clock- 
wise direction from its initial position to the left. But 

<j> = nx, 
and thus 

x' = —Rcosnx, 
y' = If. 



(21) 



In the new curve whose area may be denoted by S', 

dS' = y' dx' = y (Rn sin nx) dx, 

y sin nx.dx. (22) 

But from equation (4) 

1 C 2r 

B n = - I ysinnx dx, 

IT J0 

so that by comparison with equation (22) 

where S' is the actual area which would be indicated by the planim- 
eter whose tracing point follows the intersection of the two slots 
in the moving arms. 

In a similar manner it can be shown that 

A " = &n' CM) 

where S" is the area indicated by the planimeter when the initial 
position of the radius R is taken in the vertical position above the 
center. 

Obviously, according to equation (4), A would be determined 
by passing the planimeter over the original curve and dividing the 
area so obtained by the length of base. 



SELECTION OF THE NUMBER OF ORDINATES 45 

Assuming the apparatus to function perfectly, such a device 
would yield exact results regardless of the number of harmonics 
present in the wave. Though of interest, and of value in obtaining 
a clear idea of the theory of analysis, such a mechanical scheme is 
not very satisfactory for actual use, in that the labor required to 
obtain a complete analysis with such a device would be much 
greater than for procedures based on the measurements of a finite 
number of ordinates; particularly if tables are available to facili- 
tate the latter form of procedure. Another disadvantage of most 
mechanical analyzers is that the original curve must be drawn to 
such a scale that one period or cycle of the fundamental frequency 
has a certain definite wave length. 

37. Tables and Forms for Analysis from a Number of Ordi- 
nates. — In regard to analysis by means of a number of ordinates, 
the various schemes in use are in reality only methods for obtaining 
rapidly the numerical values of the summations indicated by the 
right hand members of equations (15) and (16). 

Of course it is possible to determine any particular coefficient, 
for instance A p , by taking every one of the t ordinates and after 
multiplying it by the cosine of p times its angle to add it with all 
similar products from the other ordinates to obtain the summation. 
This process would have to be repeated for every coefficient 
desired, and it is apparent that a great amount of time and effort 
would be consumed, particularly if many ordinates are used. 
After completing an analysis in such a manner it would be noticed 
that a great many multiplications involving the same multiplier 
and multiplicand had been performed. It is the aim of the various 
tabular schemes in use to minimize the labor of an analysis by 
eliminating as many as possible of such duplicate operations. 

38. Selection of the Number of Ordinates. Separation of 
Harmonics. — It will in general be advisable to use an even 
number of ordinates, for then the ordinates of the second half-wave 
will be located at points on the axis differing by 180 degrees from 
the abscissae used in the first half-wave. This permits of the 
separation of the odd and even harmonics, as will be pointed 
out. 

Since in a majority of cases connected with electrical engineering 
problems the periodic waves encountered will not contain the 
harmonics of even order, it is desirable to arrange an analysis 
scheme in such a way as to make possible the determination of the 



46 FOURIER'S SERIES 

odd harmonics only, but at the same time to provide means for a 
complete analysis, should such be desired. 
Since 

[— cos p/3 when p is odd 



and also 



COS p(P + ir) = , 

cos p/3 when p is even 

fa . % (— sm P/3 when p is odd 
Sinp(j8 + 7r) = \ • u - 

sin p/3 when p is even 



(25) 



it is evident that by taking the average value (one-half the alge- 
braic sum) of two ordinates separated by one-half wave length, 
or 7r, we will have the value of that portion of an ordinate which is 
due to even harmonics only. Similarly, by taking one-half the 
difference of two ordinates separated by one-half a wave length 
we obtain the ordinate due to odd harmonics only. 
Thus 

Vm" = \ \Vm + yit \~l = ordinate due to even harmonics only. (26) 
L \2 + m )\ 

Vm = h Yl)m — V/t \~| = ordinate due to odd harmonics only. (27) 

L l2 +m ;J 

This separation of the given data into two sets of data, one 
containing only even harmonics and the other only odd harmonics, 
is made possible by the use of an even number of ordinates, t. 

39. Description of Tabular Forms for Complete Analysis. — 
These forms are based on summations of the types (15) and (16), 
the summation being carried from m = to m = t — 1, instead 
of from m = 1 to m = t. The result is of course the same, but 
a simplification in notation is obtained which makes the change 
desirable. The description of the forms follows the order of use 
in making an analysis from 72 ordinates measured over a complete 
wave, but as will be apparent, it is not always necessary to use 
such a large number of ordinates. In fact the forms are arranged 
so as to accommodate an analysis for both odd and even harmonics 
from either 72, 36, or 18 ordinates per complete wave, or an 
analysis for odd harmonics only from either 36 or 18 ordinates 
measured in a half-wave. 

The given data, 72 ordinates per wave, are first separated into 
two sets of data by equations (26) and (27), one set containing 
only odd and the other only even harmonics. Form A at the end 
of the volume is used for this separation. 



DESCRIPTION OF TABULAR FORMS 47 

The 36 values of y' obtained from Form A are then to be used 
in ^orm B as original data, and from the sums and differences 
formed there the odd harmonics are determined and checked by 
Forms C and D. The process of multiplication by sine or 
cosine of pd is carried out in Form C, while in Form D a number 
of ordinates are actually computed from the resulting equation, 
which by coincidence with the original, measured ordinates, estab- 
lishes the correctness of the numerical computations. Form B is 
provided with a table for the collecting of the entire results of the 
analysis. 

The 36 values of y" covering a half wave length of the funda- 
mental frequency still remain, and from these the even harmonics 
are to be determined. This set of data is equivalent to a set of 
36 ordinates covering a complete wave length of the second har- 
monic, which may now be regarded as the fundamental frequency 
in a new set of harmonics whose frequencies are all multiples of 
two times the frequency of the original fundamental. Thus the 
even harmonics, 0, 2, 4, 6, 8, ... , etc., in the original wave 
become harmonics of order 0, 1, 2, 3, 4, ... , etc., in the new 
system whose fundamental wave length is one-half that of the 
original. 

In the new system, the even harmonics may be separated from 
the odd in exactly the same manner as was done in Form A for 
the original system, by combining ordinates separated by one-half 
of the new wave length. The entire new system is equivalent to 
a set of 36 ordinates over a complete wave, and the tables are 
therefore prepared with a notation corresponding to such a 
measurement of original data, and are described accordingly. It 
must be remembered that if it is an analysis of a new derived 
system from 72 ordinates per complete wave which is carried out, 
the actual frequencies will be twice those indicated by the tabular 
results — in other words, A 2 , Ae, Ai , Au, etc., instead of A h A 3 , 
Ai, A 7 , etc., as indicated in the tabular notation of Forms F and G. 

Form E is to be used for separating the set of 36 ordinates per 
wave length into two sets of 18 ordinates per half wave length, 
one set containing only the odd and the other only the even har- 
monics. The 18 values of y' from Form E are to be used in Forms 
F and G for the determination and checking of the odd harmonics. 

If it is known that the original wave contains no even harmonics, 
Forms F and G will serve immediately for the determination of 



48 FOURIER'S SERIES 

the odd harmonics from 18 ordinates measured over a half -wave. 
Forms B, C, and D may also be used for this purpose with the 
exception that 36 ordinates per half-wave are used, thus giving 
a greater accuracy than that obtained with 18 ordinates. After 
having utilized the 18 values of y' from Form E, the 18 values of 
y" still remain, which cover a quarter wave length of the 72 ordi- 
nate curve, or a half wave length of the derived 36 ordinate curve, 
and which are made up from the harmonics of order 4, 8, 12, 16, 
etc., of the original 72 ordinate curve, or from the harmonics of 
order 2, 4, 6, 8, etc., of the new 36 ordinate curve. 

These eighteen ordinates thus represent a full wave length of 
a periodic curve whose fundamental frequency is four times that 
of the original 72 ordinate curve, or twice that of the 36 ordinate 
curve. Forms H and I are arranged for the analysis of such a 
full wave length represented by 18 ordinates. Obviously, if the 
analysis is based upon y" from Form E, the frequency of the 
resulting terms must be multiplied by four to be put into the series 
represented by the original data of Form A or partially expressed 
by the analysis for odd harmonics in Forms B, C, and D. 

Form H as given here is substantially a reproduction of a form 
or schedule given by Grover in " Bulletin of the Bureau of Stand- 
ards," Vol. 9, page 646. The reader is referred to this bulletin for 
a partial bibliography of the subject, and for schedules or forms for 
analysis to be used with a number of ordinates still smaller than 
eighteen per complete wave, as covered by Form H. 

40. Summary. — To summarize : the forms described provide 
for a complete analysis for both odd and even harmonics including 
a constant term, from 72 ordinates per complete wave. To ac- 
complish this end, all of the forms are used — the odd harmonics 
being determined first, then those whose frequencies are twice 
those of the odd harmonics, and finally the constant term and those 
harmonics whose frequencies are four times the odd harmonic 
frequencies. 

In the use of Form I, it is of course to be noticed that twice the 
values of A and A 9 are obtained, as signified in the notation at 
the bottom of the blank form for analysis. This comes about as 
a result of equations (17) and (19), in which the factor 1/t is used 
in place of the factor 2/t. 

If desired, the odd harmonics up to and including the 35th may 
be determined from a measurement of 36 ordinates over a half 



NUMERICAL ILLUSTRATION 



49 



wave as original data, if it is sufficient to represent this half wave 
only or if it is definitely known that no even harmonics exist in the 
wave. Also, a complete analysis from 36 ordinates per full wave 
or an analysis for odd harmonics only from 18 ordinates per half 
wave is provided for, and finally an analysis for odd and even 
harmonics from 18 ordinates per full wave. 

If in any case the determination of but one particular harmonic 
is desired, it is of course possible to omit all of the computations 




X 2TT 

Fig. 10. — The Broken-line Periodic Curve Represented by Equation (28). 



except those directly involved in the various steps leading thereto. 
A little experience in the use of the forms will immediately suggest 
simplifications of procedure which may be possible in special cases. 

Analysis forms of the type shown here have been in constant 
use by the writer, and in all respects have been found the most 
satisfactory means for separating a periodic curve into its con- 
stituent harmonics. 

41. Numerical Illustration as Check on Tabular Forms. — In 
order to check the analysis forms as well as to furnish an illustration 



50 FOURIER'S SERIES 

of what might be expected in the way of accuracy from their use, 
an analysis of a curve whose constituent harmonics could be 
determined by integration was made. The curve used is made up 
of a number of straight lines whose equations are known so that 
the numerical values of the ordinates may be computed with 
accuracy. The curve is made up of the sum of two curves, one 
containing odd and the other even harmonics, so that both odd 
and even harmonics are present in the resultant. Fig. 10 shows 
the wave shape resulting from the sum of the two component 
curves. Its equation, when x is measured in radians, is 

y -y +y = ^ ~ 

*[cos(x + l)+^cos3(x+l) +^cos5(z+l)+ • • -] + |~ 

-rcos2(x-0.2) + icos6(x-0.2) + ^coslO(a;-0.2)H 1 (28) 

ir [_ y Zo J 

Expanding cos (x + a) by cos a cos x — sin a sin x, the A and 
B coefficients are obtained, and the complete curve, y = / (x), is 
represented by a Fourier's Series in which the coefficients have the 
values given in Table V. 

From the known line slopes and intercepts the 72 ordinate values 
as given in Table VI were obtained. These ordinates were equally 
spaced over a wave length. 



NUMERICAL ILLUSTRATION 



51 



TABLE V 

Coefficients of Fourier's Series Representing the Broken 
Line Curve. (Equation 28.) 



Odd harmonics 


Even harmonics 


n 


A n 


B n 


n 


A n 


B n 











2 

6 

10 


+3.141593 
-1.17273 
-0.051264 
+0.021194 




1 

3 
5 


-0.68793 
+0.14005 
-0.014447 


+1.07140 

+0.019966 

-0.048837 


-0.49581 

-0.131859 

-0.046310 


7 

9 

11 

13 


-0.019590 
+0.014322 
-0.00004657 
-0.0068366 


+0.017071 
+0.0064781 
-0.010522 
+0.0031655 


14 
18 
22 
26 


+0.025622 
+0.014096 
+0.0032343 
-0.0035298 


-0.0043209 
+0.0069560 
+0.0100132 
+0.0066560 


15 
17 
19 
21 


+0.0042990 
+0.0012120 
-0.0034871 
+0.0015814 


+0.0036798 
-0.0042356 
+0.00052861 
+0.0024155 


30 
34 • 

38 
42 


-0.0054335 
-0.0038303 
-0.00088618 
+0.0014993 


+0.0015811 
-0.0021768 
-0.0034138 
-0.0024673 


23 
25 
27 
29 
31 


+0.0012824 

-0.0020192 

+0.00051023 

+0.0011325 

-0.0012120 


-0.0020367 

-0.00026962 

+0.0016703 

-0.0010047 

-0.00053532 


46 
50 
54 
58 
62 


+0.0023463 

+0.0017093 

+0.00033942 

-0.00086036 

-0.0013066 


-0.00053648 

+0.0011082 

+0.0017132 

+0.0012457 

+0.00021941 


33 
35 
37 
39 


+0.00001552 
+0.00093928 
-0.00071187 
-0.00022322 


+0.0011691 
-0.00044503 
-0.00059852 
+0.00080679 


66 
70 
74 

78 


-0.00094224 
-0.00014212 
+0.00057232 
+0.00083222 


-0.00069223 
-0.0010296 
-0.00073312 
-0.00009020 


41 
43 
45 
47 


+0.00074782 
-0.00038225 
-0.00033030 
+0.00057196 


-0.00012015 
-0.00057277 
+0.00053501 
+0.00007122 


82 
86 
90 
94 


+0.00058318 
+0.00005418 
-0.00041517 
-0.00057568 


+0.00048332 
+0.00068647 
+0.00047218 
-0.00002855 


49 
51 
53 
55 


-0.00015940 
-0.00036330 
+0.00041626 
-0.00000931 
etc. 


-0.00050577 
+0.00032809 
+0.00017947 
-0.00042079 
etc. 


98 
102 
106 
110 


-0.00038785 
-0.00000996 
+0.00031867 
+0.00042088 
etc. 


-0.00036164 
-0.00048942 
-0.00032234 
+0.00000373 
etc. 



52 



FOURIER'S SERIES 



TABLE VI 
Seventy-two Ordinates for Check Analysis of Periodic Curve 



m 


r» 


m 


Y m 


m 


Y m 


m 


Y m 





1.40000 


18 


5.31239 


36 


2.54159 


* 

54 


3.31239 


1 


1.31274 


19 


5.57419 


37 


2.27980 


55 


3.39966 


2 


1.22546 


20 


5.83599 


38 


2.01799 


56 


3.48692 


3 


1.38540 


21 


5.85059 


39 


2.00339 


57 


3.32699 


4 


1.64720 


22 


5.76332 


40 


2.09066 


58 


3.06519 


5 


1.90899 


23 


5.67606 


41 


2.17792 


59 


2.80339 


6 


2.17080 


24 


5.58880 


42 


2.26519 


60 


2.54160 


7 


2.43260 


25 


5.42138 


43 


2.35246 


61 


2.35993 


8 


2.69439 


26 


5.15959 


44 


2.43972 


62 


2.27267 


9 


2.95620 


27 


4.89779 


45 


2.52699 


63 


2.18540 


10 


3.21799 


28 


4.63598 


46 


2.61426 


64 


2.09813 


11 


3.47979 


29 


4.37419 


47 


2.70152 


65 


2.01086 


12 


3.74160 


30 


4.11239 


48 


2.78880 


66 


1.92360 


13 


4.00339 


31 


3.85058 


49 


2.87606 


67 


1.83633 


14 


4.26519 


32 


3.58879 


50 


2.96332 


68 


1.74906 


15 


4.52699 


33 


3.32699 


51 


3.05059 


69 


1.66180 


16 


4.78879 


34 


3.06520 


52 


3.13786 


70 


1.57454 


17 


5.05059 


35 


2.80339 


53 


3.22512 


71 


1.48726 



The above table contains the ordinates for each five degrees 
along the axis of x. These ordinates represent the values to 
which the infinite series (28) would converge for the assumed 
abscissae. 

From the data given in Table VI a complete analysis was made, 
using all of the forms. The values of coefficients derived thereby 
are given in Table VII. 

The results of the complete analysis as tabulated in Table VII 
are to be compared with the corresponding values in Table V, 
which contains the true values of the coefficients. The differences 
noted arise from the effect of the disturbing terms in the sum- 
mation-equations (15) and (16). For example: the true value of 
A 2 is — 1.17273, while from the analysis we obtain A 2 = — 1.17205, 
in this case rather close agreement, but still not exact. By equa- 
tion (15), however, the value —1.17205 should be equal to the 
sum of the true values of 

A 2 + An + A 7 * + Am + Ai M + • • • • 

Forming this sum partially from Table V, 

At + A 70 + A 7i = - 1.17230, 



COMPARATIVE ACCURACY 



53 



a value nearer the result of analysis than the actual value of A 2 , 
but still differing therefrom on account of the neglected terms in 
the infinite series, 

A 14 2 + ^146 + A 2 14 + A218 + ' ' * . 

Again, by equation (16), using the true values from Table V, 

B 2 - £ 70 + B u = -0.49551, 

a quantity very nearly equal to —0.49543, which is the value of 
Z? 2 resulting from the analysis. 

TABLE VII 

Coefficients of Fourier's Series Derived by Complete Analysis 
from 72 Ordinates per Complete Wave, Fig. 10 



H 


■"» 


B n 


n 


A n 


B n 











2 


3.14159 
-1.17205 




1 


-0.68772 


1.07106 


-0.49543 


3 


0.13966 


0.01992 


4 








5 


-0.01432 


-0.04846 


6 


-0.05125 


-0.13102 


7 


-0.01930 


0.01679 


8 








9 


0.01393 


0.00633 


10 


0.02040 


-0.04578 


11 


-0.00001 


-0.01009 


12 








13 


-0.00645 


0.00294 


14 


0.02344 


-0.00912 


15 


0.00392 


0.00342 


16 








17 


0.00113 


-0.00376 


18 


0.01373 


0.00566 


19 


-0.00300 


0.00039 


20 








21 


0.00123 


0.00201 


22 


0.00419 


0.00869 


23 


0.00105 


-0.00152 


24 








25 


-0.00141 


-0.00028 


26 


-0.00153 


0.00651 


27 


0.00022 


0.00109 


28 








29 


0.00069 


-0.00045 


30 


-0.00368 


0.00334 


31 


. -0.00046 


-0.00036 


32 








33 


-0.00016 


0.00035 


34 


-0.00403 


0.00094 


35 


0.00020 


0.00011 


36 












An inspection of the values of the small coefficients of the high 
frequency terms will reveal a very considerable error in the analy- 
sis. These errors arise from the presence in the original curve of 
an infinite number of harmonics. In the analysis of any experi- 
mentally obtained curve such a condition would not be expected 
to occur, and therefore the results would not be so much in error. 
In all events, the 72 points selected on the original curve are 
represented by the resulting equation. 

42. Comparative Accuracy. — To illustrate the deviation of 
the curve obtained by analysis, from the original curve, an analysis 



54 



FOURIER'S SERIES 



of the preceding broken line curve was made from 18 ordinates 
(that is, every fourth ordinate as tabulated in Table VI was used 
in Forms H and I as original data) with the results shown in 
Table VIII. 

TABLE VIII 

Coefficients of Fourier's Series Derived by Complete Analysis 
from 18 Ordinates per Complete Wave, Fiq. 10 



n 


A n 


B n 


n 


A H 


B n 





3.15533 

-0.68939 

-1.17608 

0.14465 

0.02762 




5 
6 

7 
8 
9 


-0.02018 

-0.05493 

-0.02003 

0.01886 

0.01415 


-0.05255 


1 

2 
3 
4 


1.07510 

-0.49637 

0.01817 

0.01780 


-0.13436 
0.02705 
0.05229 







Again, for illustration, by equation (16), using true values of 
coefficients from Table V, and for t = 18 and p = 6, 

Be — Bn + -B24 — .B30 4" B& ~ B& + Bw — Bm 

-f- Z?78 — B& "T* -^96 ~~ B102 = —0.13482, 

which is nearly equal to the value —0.13436 as given by the 18 
ordinate analysis for the coefficient B 6 . 

Figure 11 shows the original broken line curve and also the 
curve represented by the finite Fourier's Series having the coeffi- 
cients given in Table VIIL The eighteen ordinates from which 
the analysis was made are shown in the figure, and corresponding 
ordinates in both curves should theoretically be equal. The 
lower, smooth curve, represented by the finite series containing 
terms up to the ninth harmonic was drawn by means of a curve 
tracing and analyzing machine designed and constructed by Pro- 
fessor J. N. LeConte, of the University of California. Small 
differences in the corresponding ordinates of the two curves may 
be attributed to slight errors in adjustment of the curve tracing 
apparatus. The approximation of the derived curve to the original 
is clearly shown, nevertheless. 

In order to facilitate the computation of ordinate values and the 
plotting of wave shapes from Fourier's Series with given coefficients 
by those who are not fortunate enough to have a curve-tracing 
device at hand, Table VIII A has been prepared, from which 



COMPARATIVE ACCURACY 



55 



the trigonometric functions of multiple angles (up to and including 
the 36th harmonic) may be obtained by finding the function of 
some angle included in the first quadrant. The upper algebraic 
sign is for cosine and the lower algebraic sign for sine functions. 






27T 



N^ 



f1\ 



2ir 



Fig. 11. — The Original Curve and the Curve Represented by the 
Results of an Eighteen Ordinate Analysis. 



TABLE VIII A 
Table for Trigonometric Functions of Multiple Angles 



ss— tea* 

Values of <t> 



tat 

■B 



Example 
23 X 75° - cos 75° 
23 X 75° - -sin 75° 





1 


2 


3 


4 


5 


6 


7 


8 


9 
































5 


5 


10 


15 


20 


25 


30 


35 


40 


45 


10 


10 


20 


30 


40 


50 


60 


70 


80 


90 


15 


15 


30 


45 


60 


75 


90 


=F75 


=F60 


T45 


20 


20 


40 


60 


80 


=F80 


=F60 


=F40 


T20 


- 


25 


25 


50 


75 


=F80 


T55 


=T=30 


T 5 


120 


I« 


30 


30 


60 


90 


=F60 


T30 


- 


130 


160 


-90 


35 


35 


,70 


=F75 


=F40 


T 5 


130 


"65 


±80 


±45 


40 


40 


80 


=F60 


T20 


~20 


160 


±80 


±40 


+ 


45 


45 


90 


=F45 


- 


Z« 


-90 


±45 


+ 


ft 


50 


50 


=F80 


T30 


~20 


170 


±60 


±10 


+40 


+90 


55 


55 


=F=70 


=F15 


~40 


±85 


±30 


ft 


+.80 


T45 


60 


60 


=F60 


- 


~69 


±60 


+ 


+.60 


T60 


- 


65 


65 


=F50 


115 


180 


±35 


+.30 


T85 


=F20 


145 


70 


70 


=F40 


130 


±80 


±10 


+.60 


=T=50 


~20 


-90 


75 


75 


=F30 


145 


±60 


ft 


+90 


T15 


~60 


±45 


80 


80 


=F20 


~60 


±40 


ft 


=F60 


120 


±80 


+ 


85 


85 


=F10 


175 


±20 


^65 


T30 


155 


±40 


+45 


90 


90 


- 


-90 


+ 


+90 


- 


-90 


+ 


+90 


95 


T85 


Z io 


±75 


+.20 


=F65 


130 


±55 


ft 


=F45 


100 


=F=80 


~20 


±60 


+40 


=F40 


160 


±20 


+.80 


- 


105 


=F75 


130 


±45 


+.60 


=F15 


-90 


+15 


T60 


I« 


110 


=F70 


~40 


±30 


^80 


Iio 


±60 


+.50 


T20 


-90 


115 


=F65 


150 


±15 


T80 


135 


±30 


> 


120 


±45 


120 


=F60 


~60 


+ 


T60 


160 


+ 


=F60 


~60 


+ 


125 


T55 


~70 


iu 


=F40 


185 


^30 


T25 


±80 


ft 


130 


T50 


^80 


+.30 


=F20 


±70 


+.60 


Iio 


±40 


+90 


135 


=F45 


-90 


ft 


- 


±45 


+90 


145 


+ 


=F45 


140 


=F40 


±80 


+.60 


120 


±20 


=F60 


180 


ft 


- 


145 


T35 


±70 


ft 


1*0 


ft 


=F30 


±65 


+.80 


1*5 


150 


=F30 


±60 


+90 


160 


^30 


- 


±30 


T60 


-90 


155 


=F25 


±50 


=F75 


180 


i 55 


130 


+ 5. 


=F20 


±45 


160 


=F20 


±40 


T60 


±80 


+.80 


~60 


i* 


~20 


+ c 


165 


=F15 


±30 


=F45 


±60 


T75 


-90 


+75 


160 


+45 


170 


=F10 


±20 


=f30 


±40 


T50 


±60 


=F70 


±80 


+90 


175 


T 5 


±10 


T15 


±20 


=F25 


±30 


T35 


±40 


=F45 


180 


- 


+ 


- 


+ 


- 


+ 


- 


+ 


- 



Page 56 



TABLE VIII A — (Continued) 
Table for Trigonometric Functions op Multiple Angles 



9^ 


10 


11 


12 


13 


14 


15 


16 


17 


18 
































5 


50 


55 


60 


65 


70 


75 


80 


85 


90 


10 


=F80 


=F70 


=F60 


=P50 


=F40 


=F30 


=F20 


T10 


- 


15 


T30 


T15 


- 


115 


130 


145 


160 


175 


-90 


20 


~20 


~40 


160 


180 


±80 


±60 


±40 


±20 


+ 


25 


~70 


±85 


±60 


±35 


±10 


t» 


+.40 


+.65 


+90 


30 


±60 


±30 


+ 


+.30 


+.60 


+90 


T60 


T30 


- 


35 


±10 


+.25 


+60 


=F85 


T50 


T15 


120 


155 


-90 


40 


+.40 


+.80 


T60 


=F20 


120 


160 


±80 


±40 


+ 


45 


+90 


=F45 


- 


~45 


-90 


±45 


+ 


+45 


+90 


50 


T40 


Iio 


~60 


±70 


±20 


+.30 


+.80 


=F50 


- 


55 


1 10 


~«B 


±60 


± 5 


+.50 


T75 


=F20 


135 


-90 


60 


~60 


±60 


+ 


+60 


=F60 


- 


160 


±60 


+ 


65 


±70 


±5 


+.60 


T55 


1 10 


175 


±40 


+25 


+90 


70 


±20 


+50 


=F60 


Iio 


180 


±30 


+.40 


=F70 


- 


75 


+.30 


T75 


- 


175 


±30 


+45 


=F60 


115 


-90 


80 


+.80 


=F20 


~60 


±40 


+«> 


=F60 


120 


±80 


+ 


85 


=F50 


135 


±60 


+.25 


=F70 


115 


±80 


+ 5 


+90 


90 


- 


-90 


+ 


+90 


- 


-90 


+ 


+90 


- 


95 


~50 


±35 


+.60 


T25 


170 


±15 


+.80 


=F 5 


-90 


100 


±80 


+a> 


=F60 


~40 


±40 


+.60 


=F20 


180 


+ 


105 


±30 


+75 


- 


±75 


+.30 


T45 


160 


±15 


+90 


110 


£20 


=F50 


~60 


±10 


=F80 


130 


±40 


+.70 


- 


115 


+.70 


~ 5 


±60 


+.55 


=F10 


±75 


+40 


T25 


-90 


120 


T60 


~60 


+ 


=F60 


Zoo 


+ 


=F60 


Zoo 


+ 


125 


=F10 


±65 


+60 


I 5 


±50 


> 


120 


±35 


+90 


130 


~40 


±10 


=P60 


~70 


+20 


=F30 


±80 


+.50 


- 


135 


-90 


+.45 


- 


±45 


+90 


~45 


+ 


=F45 


-90 


140 


±40 


=F80 


160 


+20 


T20 


±60 


+.80 


140 


+ 


145 


> 


=F25. 


±60 


+85 


1 50 


+15 


=F20 


±55 


+90 


150 


+.60 


~30 


+ 


=F30 


±60 


+90 


160 


+30 


- 


155 


=F70 


~85 


+.60 


135 


t„ 


T15 


±40 


T65 


-90 


160 


=F20 


±40 


=F60 


±80 


+.80 


160 


+ 40 


~20 


+ 


165 


130 


+15 


- 


±15 


T30 


±45 


=F60 


±75 


+90 


170 


180 


+.70 


160 


+^50 


"40 


+.30 


120 


t» 


- 


175 


±50 


=F55 


±60 


=F65 


±70 


=F75 


±80 


T85 


-90 


180 


+ 


- 


+ 


- 


+ 


- 


+ 


- 


+ 



Page 57 



TABLE VIII A — (Continued) 
Table for Trigonometric Functions of Multiple Angles 



^v? 

• ^ 


19 


20 


21 


22 


23 


24 


25 


26 


27 
































5 


T85 


=F80 


T75 


=F70 


+65 


T60 


=F55 


=F50 


T45 


10 


Zio 


120 


130 


1*0 


150 


~60 


~70 


Z80 


-90 


15 


±75 


±60 


±45 


±30 


±15 


+ 


f 


+J30 


%m 


20 


> 


p. 


+.60 


+.80 


T80 


TOO 


=F40 


=F20 


- 


25 


=F65 


=F40 


=F15 


Z io 


135 


Zoo 


Z85 


±70 


±45 


30 


130 


160 


-90 


±60 


±30 


+ o 


+.30 


+.60 


+90 


35 


±55 


±20 


> 


^50 


+"85 


=F60 


=F25 


Zio 


Z« 


40 
45 


=f=45 


+.80 
- 


=F60 
~45 


=F20 
-90 


~20 
±45 


~60 

+ 


±80 

> 


±40 
+90 


+ 
=F45 


50 


150 


±80 


±30 


^20 


+ 70 


=F60 


=F10 


140 


-90 


55 


±35 


+.20 


p, 


T50 


Z 5 


loo 


±65 


±10 


*■ 


60 


+.eo 


T60 


- 


~60 


±60 


+ 


+.60 


=F60 


- 


65 


=F25 


1*0 


±75 


±10 


+.55 


T60 


Z 5 


Z70 


±45 


70 


~70 


±40 


+.30 


=F80 


T10 


Zoo 


±50 


+.20 


+90 


75 


±15 


+.60 


=F45 


130 


±75 


+ 


+75 


=F30 


^45 


80 


+.80 


=F20 


160 


±40 


+ 40 


=F60 


~20 


±80 


+ o 


85 


=F 5 


180 


±15 


+.70 


=F25 


Zoo 


±35 


+.50 


T45 


go 


-90 


+ 


+90 


- 


-90 


+ 


+90 


- 


-90 


95 


t« 


=F80 


Z 1 * 


±70 


^25 


=F60 


Z35 


±50 


> 


100 


=F80 


120 


±60 


t.40 


T40 


Zoo 


±20 


+.80 


- 


105 


Zu 


±60 


t" 


T30 


~75 


+ 


=F76 


Z30 


±45 


110 


±70 


*■ 


=F30 


"80 


p. 


=F60 


Z50 


±20 


+90 


115 


+25 


=F40 


175 


> 


=F55 


Zoo 


± 5 


=F70 


145 


120 


=T=60 


ZOO 


+ 


=F60 


160 


+ o 


T60 


~60 


+ 


125 


135 


±20 


=F75 


150 


± 5 


=F60 


~65 


+,. 


=F45 


130 


±50 


^80 


130 


±20 


=F70 


Zoo 


> 


=F40 


-90 


135 


+45 


- 


±45 


+90 


Z<8 


+ 


=F45 


-90 


+45 


140 


=F40 


±80 


+.60 


~20 


±20 


=F60 


^80 


+ 40 


- 


145 


155 


+.20 


=F15 


±50 


T85 


Zoo 


+J25 


T10 


±45 


150 


±30 


=F60 


-90 


+.60 


130 


+ 


=F30 


±60 


+90 


155 


+.65 


"40 


> 


=F10 


±35 


T60 


±85 


+.70 


"45 


160 


=F20 


±40 


=F60 


±80 


+.80 


Zoo 


+ 40 


"20 


+ 


165 


175 


+.60 


~45 


^30 


in 


+ 


T15 


±30 


=F45 


170 


±10 


=F20 


±30 


=F40 


±50 


T60 


±70 


=F80 


-90 


175 


> 


180 


> 


170 


+.65 


Zoo 


+65 


~50 


> 


180 


- 


+ o 


- 


+ 


- 


+ 


- 


+ o 


- 



Page 58 



TABLE VIII A— (Concluded) 
Table for Trigonometric Functions of Multiple Angles 



8 


28 


29 


30 


31 


32 


33 


34 


35 


36 
































5 


=F40 


=F35 


=F30 


=F25 


T20 


=F15 


=F10 


T 5 


-0 


10 


±80 


±70 


±60 


±50 


±40 


±30 


±20 


±10 


+0 


15 


+«) 


p. 


+90 


=F75 


T60 


=F45 


=F30 


T15 


-0 


20 


~20 


_40 


160 


180 


±80 


±60 


±40 


±20 


+0 


25 


±20 


+ 5 
+ 6 


+30 


+.55 


+.80 


=F75 


=F50 


=F25 


-0 


30 


=F60 


=F30 


- 


"30 


_ 60 


-90 


±60 


±30 


+0 


35 


180 


±65 


±30 


I' 


> 


P> 


T70 


=F35 


-0 


40 


+.40 


+.80 


=F60 


=F20 


120 


~60 


±80 


±40 


+0 


45 


- 


~45 


-90 


±45 


+ 


+45 


+90 


=F45 


-0 


50 


±40 


t» 


+.60 


=F70 


=TF20 


~30 


180 


±50 


+0 


55 


=F80 


=F25 


130 


185 


±40 


+15 


+.70 


T55 


-0 


60 


160 


±60 


+ 


+« 


=F60 


- 


160 


±60 


+0 


65 


+.20 


+.85 


=F30 


~35 


±80 


±15 


+.50 


=F65 


-0 


70 


_T20 


~50 


±60 


+10 


+.80 


=F30 


~40 


±70 


+0 


75 


±60 


+15 


+90 


=F15 


^60 


±45 


+.30 


=F75 


-0 


80 


+.80 


=F20 


160 


±40 


+.40 


=F60 


~20 


±80 


+0 


85 


1*0 


±55 


+.30 


=F65 


120 


±75 


+10 


=F85 ' 


-0 


90 


+ 


+90 


- 


-90 


+ 


+90 


- 


-90 


+0 


95 


=F40 


155 


±30 


+.65 


=F20 


175 


±10 


+.85 


-0 


100 


±80 


+.20 


=F60 


~40 


±40 


+.60 


=F20 


~80 


+0 


105 


+.60 


=F15 


-90 


+15 


=F60 


~45 


±30 


+75 


-0 


110 


~20 


±50 


+.60 


=F10 


±80 


+M> 


=F40 


170 


+0 


115 


±20 


T85 


130 


±35 


+.80 


115 


±50 


+.65 


-0 


120 


=F60 


~60 


+ 


=F60 


160 


+ 


=F60 


~60 


+0 


125 


~80 


+25 


=F30 


±85 


+.40 


T15 


±70 


+55 


-0 


130 


+.40 


=T=10 


±60 


+70 


~20 


±30 


=F80 


"50 


+0 


135 


- 


±45 


+90 


~45 


+ 


T45 


-90 


+45 


-0 


140 


±40 


T80 


160 


+.20 


=F20 


±60 


^80 


140 


+0 


145 


=F80 


165 


+.30 


T 5 


±40 


T75 


170 


+.35 


-0 


150 


160 


+.30 


- 


±30 


=F60 


-90 


+0) 


~30 


+0 


155 


+.20 


=F 5 


±30 


=F55 


±80 


+75 


~~50 


+.25 


-0 


160 


T20 


±40 


=F60 


±80 


+80 


~60 


+40 


~20 


+0 


165 


±60 


T75 


-90 


p. 


- 60 


> 


130 


t« 


-0 


170. 


+.80 


~70 


^60 


_50 


t" 


~30 


+.20 


10 


+0 


175 


^40 


+.35 


130 


^25 


~20 


> 


Iio 


l> 


-0 


180 


+ 


- 


+ 


- 


+ 


- 


+ o 


- 


+0 



Page 59 



CHAPTER IV 

TREATMENT OF NON-SINUSOIDAL ALTERNATING 
QUANTITIES. THE USE OF THE OSCILLOGRAPH 

43. General. — Following out the plan, as stated, of consider- 
ing some of the more important matters relating to special in- 
vestigation of transmission line phenomena, this chapter, which 
deals with the methods of obtaining the periodic curve and of 
utilizing the results of harmonic analysis of the same, is included. 

For a large part of alternating current investigation the complex 
quantity method of treatment, to which such prominence has been 
given by the work of Steinmetz, is the ideal one. A very brief 
outline of the method is sufficient here since detailed treatment is 
to be found in most all of the modern texts. 

The simplest alternating quantity may be described as one 
which varies harmonically with time between numerically equal 
positive and negative limits. Such a variation is represented 
algebraically by a trigonometric function — either cosine or sine, 
or both, depending upon the arbitrarily selected instant from which 
to count time. Thus, the displacement of a clock pendulum 
(whose motion is approximately harmonic) from its position of 
equilibrium would be represented by a cosine function of time if 
the zero of time was chosen as the moment when the pendulum 
was in an extreme position away from its equilibrium position; 
while if the zero of time was taken at the instant the pendulum 
passes through its equilibrium position, the displacement there- 
from is represented by a sine function of time. This applies equally 
to alternating electrical quantities, in that we may assign to one 
of the quantities in the system either a cosine or a sine (or a com- 
bination of both) variation by the proper selection of our arbitrary 
zero of time. All other quantities must be referred to this same 
zero, and thus such an arbitrary selection of the particular trigono- 
metric function for one of the quantities can be made for this one 
quantity only — all others being represented by such functions 
as will give to them their actual phase displacement from the 

60 



THE POLAR DIAGRAM 



61 



initial quantity. By properly selecting the zero of time important 
simplifications may often be made in the form of the analytic 
expressions for physical phenomena. 

44. The Polar Diagram and Vector Representation of Alter- 
nating Quantities. — Let the alternating current flowing in an 
electrical circuit be represented by 

i = a cos cat. (1) 

Plotted in rectangular coordinates there results the familiar 
cosine curve. Plotted in polar coordinates with OX taken as the 



.- ^js 




Fig. 12. — The Circles Representing Trigonometric Functions Plotted 
in Polar Coordinates. OX is the Reference Line. 



reference line (Fig. 12) and with the angle 6 = <at as independent 
variable counted positive in the clockwise direction, the curve 
becomes a circle whose diameter lies along the reference line OX 
and has a length a. This arbitrary selection of positive direction 
of rotation is opposite to that usually used in mathematical dis- 
cussions. It is used thus in order to lead to a standardized form 
of expression for impedances. 

Suppose the current i to flow through a circuit containing 



62 NON-SINUSOIDAL ALTERNATING QUANTITIES 

resistance, inductance, and capacity; r, L, and C, respectively. 
Then by the fundamental laws of the electric circuit, the impressed 
e.m.f. at any instant is 

e = ri + Lj t + -gfi dt, (2) 

which, for a current flow represented by equation (1), becomes 

e = ar cos ut — a(uL ^jsinut. (3) 

The e.m.f. thus consists of two components, a cosine and a sine 
component when referred to the same zero instant as the current 
i. The two dotted circles in Fig. 12 represent these two compo- 
nents of e, where 

e = ei + e 2 , 
ei = ar cos cat, 



fy = — a ( laL -p, J sin at. 



(4) 



The single curve representing e as the resultant or sum of the 
two circles, e x and e 2 , has its diameter displaced from the horizontal 
by an angle whose tangent is 



uL ^ 



Fig. 13 shows the two circles, e and i. 

For convenience, the circles may be omitted from the diagram 
and the diameters only, retained. Further, since, in general, 
effective or root-mean-square values are desired, the diameters 
may be drawn to such a scale as to represent these values which 

are —7= times the maximum. Such a representation of alternating 

v2 
quantities constitutes the so-called "vector diagrams." They are 
in no sense vector diagrams, however, since the lines drawn therein 
are, in effect, only the diameters of the polar circles which have 
arbitrarily been omitted, and do not possess any of the properties 
of a physical "vector." The only point of resemblance lies in the 
fact that the parallelogram law of combination of such diameters 
holds true for different components of the same alternating quan- 



COMPLEX QUANTITY REPRESENTATION 



63 



tity. Since the term "vector diagram" has become so familiar 
and receives, such wide usage, it appears proper to retain it here, 
the distinction between such diagrams and the true vector diagram 
of mechanics being recognized, however. 




Fig. 13. — The Polar Circles Representing Voltage and Current. 



45. Complex Quantity Representation. — Having reduced the 
representation of trigonometric functions to the simple straight 
lines of the vector diagram, further simplification is to be made. 
The positions of the ends of the vectors in the diagram may be 
specified in a system of rectangular coordinates, and the diameters 
of the polar circles which are represented by these vectors are 
completely located thereby. In naming the coordinates of the 
end of the vector, horizontal distances are taken positive to the 
right and negative to the left of the origin. Distances in a per- 
pendicular direction are positive when above and negative when 
below the horizontal axis. To distinguish these latter components 
from the horizontal, the prefix j is used. 

According to this convention, the two alternating quantities, 
i and e, as shown in Fig. 13 by their polar circles and in Fig. 14 by 



64 



NON-SINUSOIDAL ALTERNATING QUANTITIES 



the corresponding vectors, and whose equations in time are given 
by (1) and (3), may be most conveniently expressed by 

In (5), capital letters are used, indicating an alternating quantity 
specified in vector notation. Cosine functions are always repre- 



(5) 




Fig. 14. — The Vectors Representing the Diameters of the Polar Circles. 



sented by the term not affected with the symbol j. A sine wave 
of the form 



becomes 



fy = —a[ o)L 7: ) sin wt 

27 2 =+ja(o,L-^) 



in the new notation. 

Negative sine functions take the +j as prefix, and positive sine 
functions the — j, while the cosine functions require no prefix and 
retain their original algebraic sign when expressed in the new 
notation. 

Alternating quantities are thus expressible in either of the two 
analytical forms — by trigonometric functions, or by means of 
the simplified notation above. The vector C = a + jb may 



THE COMPLEX OPERATOR 65 

immediately be written c = a cos cot — b sin oit , unless effective 
values are represented by the vector notation, in which case, to 
give true instantaneous values, we must write 

c = V2 (a cos at — b sin cat). 

46. The Complex Operator — Ratio between Two Alternating 
Quantities Expressed as Vectors. — The method of expression 
of alternating quantities by the use of so-called vectors, and the 
simplified notation for such, involving the use of the prefix j, does 
not imply that any significance be attached to the symbol j other 
than that of a mere distinguishing mark to designate a distance 
above or below the reference axis in the polar diagram. Great use 
is made, however, of a further assigned significance. 

In equation (5) the ratio between e.m.f. E and current I may 
be formed, E and I retaining their vector expression. 

ar + jatooL - — ^j 

I a + jO « 

serves as a defining equation for the quantity Z, which must have 
the same physical dimensions as electrical resistance, being the 
ratio of voltage to current. By division in (6) 

This quantity Z cannot represent any alternating quantity of a 
type similar to E or 7, for if the ratio of instantaneous values is 
taken 



at — a ( ojL -pz I sin ojt . ., . 

V wC/ = r - ( coL — J 

a cos oit \ oiC/ 



ar cos 

" tan oit 
1 



and this ratio passes recurrently through positive and negative 
infinite values, while writing (7) in a trigonometric form by the 
transformation defined in the preceding paragraph yields 

r cos oit — (oiL ~ ) sin oot 



t(— a 



which is a totally different result, and therefore wrong. 

The quantity Z as defined by (7) is merely an operator. Multi- 
plication of the current value I by this operator yields the vector 



66 NON-SINUSOIDAL ALTERNATING QUANTITIES 

expression for e.m.f., or inversely, division of the voltage expres- 
sion E, in vector form, by this operator yields the expression for 
current, 7. To be such an operator, this quantity Z must be in- 
dependent of the arbitrary selection of the zero instant of time. 
This property may be made the basis for an interpretation of the 
significance of the symbol j. 

Take the origin of time so that the expression for current as 
given in equation (1) becomes 

i = a sin <at. 
Then 

e = ri + L Jt + lj idt 
= ar sin wt + a ( coL ~ J cos at, 

and in the vector notation 

l = 0-ja, 

from which the ratio 

aLL--^)-jra 

Z= | = _V «£/ (8) 

I -ja 

Equating the values of Z as given by (7) and (8), 

{" L -^c)- jr 



= r+j 



( wL -^) 1 



-J 
from which, after multiplying throughout by — j, 

-f = 1, or j = \/^I. (9) 

The assignment of this numerical significance to the quantity 
j does not prevent its use as a designating symbol for the upward 
direction in the vector diagram, while it does enable all formal 
algebraic operations, multiplication, addition, extraction of roots, 
etc., incident to computations relative to complicated electrical 
systems, involving the use of such operators as the complex 
quantity Z, or its reciprocal Y, to be rigorously carried out. The 
reader will find full discussion of the use of such complex quantity 



LIMITATIONS OF THE COMPLEX QUANTITY METHOD 67 

representation of impedances, Z, in connection with the vector 
representation of alternating electrical quantities in any treatise 
on alternating currents — particularly in Steinmetz' "Alternating 
Current Phenomena," where the method is developed in great 
detail and applied to a large number of special cases. 

47. Transformation from Trigonometric Functions of Time to 
Complex Representation. — The points for which emphasis is 
sought here are: 1st, the ready transformation from a trigono- 
metric expression for an alternating quantity, a cos cot -\- b sin cot, 
to a vector representation, a — jb, and vice versa; and 2nd, the 
fundamental difference between the vectors used in the diagram 
which represent alternating quantities, and the ratios between 
such quantities which are the complex operators and can in no 
sense be drawn in such a vector diagram, although they are repre- 
sented algebraically by a similar notation. Thus E = e\ + je 2 
represents a definite alternating quantity, 

ei cos cot — e 2 sin cot, 

and as such may be drawn as a vector in the conventional vector 
diagram, ei and e 2 being the coordinates of the end of the vector, 
while Z = r + jx represents only the complex operator which 
signifies the ratio between voltage impressed upon and current 
flowing in a circuit of resistance r and inductive reactance x. 

For discussion of the quantities, resistance, reactance, conduct- 
ance, and susceptance, as well as the allied quantities, impedance 
and admittance, the reader is referred to any text on alternating 
currents. 

48. Superposition. Use and Limitations of the Complex 
Quantity Method. — In any electrical circuit whose physical 
properties, resistance, self-inductance, capacity, etc., remain con- 
stant regardless of the magnitudes of the voltages or currents 
present, any number of impressed voltages, if they be of the same 
frequency and pure sinusoidal wave shape, will produce currents 
and potential differences among the several portions of the system, 
all of which are of the same frequency and of a pure sinusoidal 
form. Calculations of phenomena in such circuits are made using 
the vector notation for alternating quantities and the complex 
form for the operators which represent the properties of the various 
portions of the circuit (impedance, admittance, etc.), in exactly 
the same manner as would be done for a similar combination of 



68 NON-SINUSOIDAL ALTERNATING QUANTITIES 

resistances only under the influence of unvarying, unidirectional 
impressed voltages. Kirchhoff's Laws, with impedance as the 
generalized term substituted for resistance, form the basis for the 
development of equations. 

If, in such a system in operation, a source of e.m.f. of different 
frequency be inserted, the calculations relative to the phenomena 
of the existing frequency still hold true, but in addition to these 
phenomena there is added a new set of the same frequency as the 
new e.m.f. whose frequency differs from the other. In other 
words, the method of superposition is to be employed in making 
calculations, and, in fact, the phenomena may themselves be re- 
garded as the resultant of several component parts. It must be 
remembered that the scheme of superposition is merely a con- 
ceptual one, however, and that at any instant there is but one 
actual current or potential difference present in a given portion of 
a system. It is necessary to realize that not only is the method 
of superposition, but also the use of vector representation inap- 
plicable, when the properties of a circuit vary with variations in 
the magnitude of currents or voltages. A case in point is that of 
an iron cored reactance, where the permeability of the core, and 
thereby the self-inductance of the circuit when defined in any way 
whatsoever, is not a constant, but varies with the magnetizing 
force. In such cases, sinusoidal impressed e.m.f.'s lead to non- 
sinusoidal currents, and vice versa. 

As opposed to the above limitations may be mentioned the case 
in which the apparent constants or properties of a circuit vary with 
the changes in frequency, but not with changes in magnitudes. 
Such cases may be rigorously treated by the complex quantity 
(vectors and complex operators) method provided care is used in 
selecting the values of the circuit properties which exist at the 
frequency under discussion. For example, if a circuit contains an 
air-cored inductance coil, it would be perfectly proper to base 
vector quantity calculations for this circuit on the values of effec- 
tive resistance and inductance of this coil at the frequency in 
question. In general the effective resistance and effective in- 
ductance of a compactly wound coil will change with change in 
frequency, but these changes arise from the fact that the coil to 
which we attach the notion of simple resistance and self-inductance 
really consists of a very intricate combination of inductances, 
resistances, and capacities (capacities between adjacent turns) and 



NON-SINUSOIDAL ALTERNATING QUANTITIES 



69 



that the conductor of which it is composed is of finite size and thus 
has an unequal current distribution over its cross section, so that 
from the very nature of the case we could not expect the apparent 
resistance and self-inductance to be the same at different fre- 
quencies. By regarding such a coil as an infinitely complicated 
system of simple parts of circuits we would still expect the proper- 
ties of the coil to be independent of the magnitude of current flow; 
and such is the case, the effective constants determined for any 
particular frequency and current strength holding true for all 
values of current at this same frequency. In such a case the only 
significance the terms "effective resistance" and "effective self- 
inductance" can have are respectively the real portion and - 
times the reactance, or j term, of the observed complex ratio 



WWV^WW 



•-AAAAA- 



>Vib. 



L =0.00140 henrys. 
r =0.200 ohms. 
C'=10xl0" 6 farads. 
r'=30.0ohms. 



Fig. 15. — Oscillograph Distortion Circuit for Current Measurements. 



between impressed voltage and current at the particular frequency 
used. 

49. Numerical Example of Complex Quantity Treatment of 
Non-Sinusoidal Alternating Quantities. — To illustrate the 
method of treatment of non-sinusoidal periodic alternating 
currents by the vector method, application to some special cases 
will be shown. 

Suppose that in order to determine the wave-shape of the 
current flowing in a given conductor an oscillograph is used, 
connected as shown in Fig. 15. In this figure i is the current in 
the main conductor, whose wave shape is to be determined. The 
oscillograph galvanometer is connected in series with a resistance 



70 NON-SINUSOIDAL ALTERNATING QUANTITIES 

r' and a condenser C", the whole being shunted around an induc- 
tance coil of resistance r and self-inductance L. The constants 
of the two circuits are adjusted so that the proper amount of cur- 
rent for the operation of the oscillograph galvanometer is diverted 
through the same. Reasons for selecting a circuit arrangement as 
shown will be given later. The upper curve in Fig. 16 shows the 
wave shape as recorded by the oscillogram. By means of a direct 
current calibration of the galvanometer, the amount of current per 
unit linear deflection of the spot of light from the zero-point may 
be determined, and thus the scale of ordinates may be such as to 
indicate directly the number of amperes of current flowing in the 
oscillograph vibrator circuit. The y coordinates of the upper 
curve in Fig. 16 give instantaneous values of the current i'. 

Assuming for the present that the oscillograph records accurately 
the instantaneous value of current flowing therein, correction must 
be made for the distorting effect of the circuit arrangement in order 
to determine the actual wave shape of the current in the main 
circuit, since it is only when non-inductive resistances are used, 
both for the shunt and for the vibrator circuit proper, that the 
wave shape as shown by the oscillogram will be the same as that 
of the line current. 

Analyzing the oscillogram for i ' by means of Analysis Forms F 
and G, from 18 ordinates measured over one-half wave, and with 
the origin taken as indicated in the figure, the equation of the 
vibrator current was determined to be 

i> = -0.0204 cos + 0.0347 sin 

+0.0210 cos 3 + 0.0018 sin 3 

-0.0034 cos 5 - 0.0072 sin 5 

-0.0038 cos 7 + 0.0037 sin 7 

-0.0052 cos 9 - 0.0002 sin 9 

-0.0003 cos 11 + 0.0058 sin 11 

+0.0001 cos 13 - 0.0003 sin 13 

+0.0004 cos 15 - 0.0003 sin 15 
, - 0.0004 cos 17 - 0.0005 sin 17 0. (10) 

The fundamental frequency is 60 cycles per second. Then, 
co„ = 2 irn (60) = 377 n = angular velocity for nth harmonic. If 

Z„ = r + joj n L = impedance of shunt, 



NON-SINUSOIDAL ALTERNATING QUANTITIES 



71 



and 



Z n ' = r' — j — p^ = impedance of vibrator circuit, 



by the application of Kirchhoff's Law it is easily shown that in 
vector notation 



1 + §Ai 



(ID 



The vector expressions for vibrator current may be written 
immediately from equation (10) ; thus, for example, using maximum 
values, 

W = 0.0210 -j 0.0018, 

while the term in parentheses in (11) is to be computed from the 
values of the circuit constants. The following table indicates the 
main features and results of the calculation by equation (11). 

TABLE IX 

Correction for Oscillograph Distortion Circuit 

(Maximum values of current) 



n 


z n ' 


Z n 


(' + fc') 


/»' 


In 


1 


30-/265 


0.20 +/0. 528 


-419 -/215 


-0.0204-/0.0347 


1.10+/18.9 


3 


30-/88.3 


0.20+/ 1.58 


-51.8-/25.6 


0.0210-/0.0018 


-1.13-/0.44 


5 


30-/53.0 


0.20+/2.64 


-18.1-/12.8 


-0.0034+/ 0.0072 


0.154-/0.087 


7 


30-/37.9 


0.20+/3.70 


-8.76-/8.64 


-0.0038-/0.0037 


0.001 +/0.065 


9 


30-/29.4 


0.20+/4.75 


-4.93-/6.55 


-0.0052+/ 0.0002 


0.027 +/0. 033 


11 


30-/24.1 


0.20+/5.81 


-2.97-/5.30 


-0.0003-/0.0058 


-0.030 +/0. 019 


13 


30-/20.4 


0.20+/6.86 


-1.84-/4.46 


0.0001 +/0. 0003 


0.001-/0.001 


15 


30-/17.7 


0.20+/7.92 


-1.14-/3.84 


0.0004 +/0. 0003 


0.001-/0.002 


17 


30-/15.6 


0.20+/8.98 


-0.66-/3.37 


-0.0004+/ 0.0005 


0.002 +/0. 001 



The complex values of I n give the following equation which 
represents the line current i as a function of time when referred 
to the same zero as the oscillograph current i' . 

i= 1.10 cos 6 - 18.9 sin 9 
-1.13cos30 + O.44sin30 
+0.154 cos 5 + 0.087 sin 5 
+0.001 cos 7 - 0.065 sin 7 
+0.027 cos 9 - 0.033 sin 9 
-0.030 cos 11 - 0.019 sin 11 
+0.001 cos 13 + 0.001 sin 13 
+0.001 cos 15 + 0.002 sin 15 
+0.002 cos 17 - 0.001 sin 17 0. (12) 



72 



THE USE OF THE OSCILLOGRAPH 




-24 



Fig. 16. — The Distortion Produced by the Circuit of Fig. 15. Upper: Oscil- 
lograph Vibrator Current, i'. Lower: Main Line Current, i, by Analysis. 



The lower curve in Fig. 16 shows the wave represented by 
equation (12), in the proper phase position with reference to 
the upper curve which represents the oscillograph vibrator 
current. 



DISTORTING CIRCUITS FOR USE WITH OSCILLOGRAPH 73 

A comparison of the two curves, i and i', should immediately 
show the advantage of using a circuit connection as illustrated by- 
Fig. 15. The higher harmonics are very much accentuated in the 
vibrator circuit, and thus in the resulting oscillogram, thereby 
rendering their determination much more accurate than would be 
possible if an undistorted wave shape were used. The amount 
of the distortion is strikingly illustrated by a comparison of the 

Z' 

different frequency values of the quantity 1 + ■=-, as given in 

Zi 

Table IX, and which represent the ratios of line current to oscillo- 
graph current. 

50. Distorting Circuits for Use with the Oscillograph. — The 
oscillograph is used to obtain one of two things — either the vave 
shape of the current in some conductor, or the wave shape of a 
potential difference or an electromotive force. These, at any rate, 
will be the quantities of particular interest in a consideration of 
transmission phenomena. With the exception of an electrostatic 
form of oscillograph, all types of oscillographs require for their 
operation a current of appreciable magnitude, since they are based 
upon either an electromagnetic or hot-wire principle, and thus 
they cannot indicate potential differences directly but only through 
the medium of a current produced thereby. One of the most 
popular commercial oscillographs requires a current of approxi- 
mately 50 milliamperes to operate it satisfactorily, so that in 
many instances the effect of a current flow of this magnitude on 
the phenomena in the system under investigation cannot be 
neglected. For the recording of potential differences, the electro- 
static form of instrument is theoretically the ideal one — its 
electrostatic capacity being so small that the current consumed 
thereby is negligible, except at frequencies far higher than the 
vibrating system could record. The electromagnetic type, on 
the other hand, affords almost an ideal instrument for the indica- 
tion of current, since the resistance of the galvanometer circuit may 
be made as low as one or two ohms, and which, for the indication 
of currents of large magnitude, could be connected around a shunt 
of very low resistance. The whole combination would thus cause 
a drop in potential of only one-twentieth to one-tenth of a volt, 
a quantity usually negligible. 

In a large number of instances where it is definitely known 
that the effect of the oscillograph circuit in altering the ex- 



74 THE USE OF THE OSCILLOGRAPH 

isting phenomena in the system can be neglected, or when it is 
desired to record the phenomena which occur as affected by 
the oscillograph regardless of whether or not they are the same as 
would occur with the oscillograph removed, a very great increase 
in the accuracy of determination of the various' frequency com- 
ponents can be secured by so arranging the oscillograph circuit 
that all of the harmonics whose values are desired produce cur- 
rents through the galvanometer of the same order of magnitude. 
In the numerical example just considered this condition was 
secured by means of the circuit combination shown in Fig. 15. 
With such a circuit arrangement any particular harmonic may be 
accentuated in the oscillogram, for by adjusting the values of L 
and C a condition approximating current resonance for this 
particular frequency may be secured, in which case the vibrator 
current may be made much greater, even, than the total line 
current. In order to avoid difficulties from current resonance 
where such is not desired, it is always well to insert considerable 
resistance in series with the galvanometer and condenser, and 
while observing the wave shape on the visual screen to reduce this 
resistance until the desired distortion is obtained. If one fre- 
quency predominates to an undesirable extent, a change in the 
condenser capacity will usually remedy the difficulty. In using 
any oscillograph circuit containing capacity, it is well to always 
make the first connection to the circuit under test through a 
comparatively high resistance which subsequently is gradually 
removed. This prevents a transient flow of current, into the 
condenser, of a magnitude sufficient to burn out the protective 
fuses placed in series with the galvanometer. 

In this discussion of distorting circuits to be used for accentuat- 
ing the higher harmonics, it is of course to be understood that the 
scheme is to be applied only when the system is operating in a 
steady condition. It is obvious that an investigation of transient 
phenomena would not permit of the use of circuits of the type 
discussed here. 

If the extreme distortion possible with the connection shown in 
Fig. 15 is not desired, a very convenient connection consists in 
shunting an inductance coil with a non-inductive vibrator circuit 
— that is, using the connection in Fig. 15 with the condenser C 
short-circuited. Another connection which would yield approxi- 
mately the same result would be that in which the galvanometer 



DISTORTING CIRCUITS FOR USE WITH OSCILLOGRAPH 75 

circuit containing a condenser and resistance was shunted around 
a non-inductive resistance. 

In making an oscillogram of the current flowing in a high 
voltage system, where it would be impracticable to utilize any- 
direct shunt connection on account of the large potential differ- 
ences, most satisfactory results may be obtained by using a mutual 
inductance coupling between the high voltage circuit and the 
oscillograph circuit.* For this purpose, air-core inductance coils 
of large diameters and small winding sections will permit of a fairly 
close electromagnetic coupling with a separation between coils 



vvwWIif 

M 

mu\ 



l-AAAAA 



iVib. 



Fig. 17. — Electromagnetically Coupled Current-distorting Circuit. 



great enough to eliminate any danger from the high potential 
system. One of the coils is connected in series with the high 
voltage circuit and the other is connected in series with the oscillo- 
graph vibrator and the proper regulating resistance. If great 
distortion is desired, the oscillograph circuit may contain a con- 
denser. For a circuit connection as shown in Fig. 17 the following 
equation expresses the relation between oscillograph and line cur- 
rents, from which the results may be corrected so as to yield the 
true wave shape of line current. 



n — Jl/f -* n • 



(13) 



* The use of current transformers for this purpose is not recommended if 
a distortion circuit is to be employed, for the impedance of such a circuit may 
require a voltage sufficient to prevent a proper current transformation. 



76 THE USE OF THE OSCILLOGRAPH 

Equation (13) is to be used in exactly the same manner as 
equation (11), illustration of which has been given. If the con- 
denser is not present, that is, equivalent to being short-circuited, 
the term l/w n C" is zero, in (13). 

If the wave shape only of the line current is desired, and not the 
absolute magnitude, it is obvious from (13) that the value of the 
mutual inductance, M, need not be known in order to make the 
reduction. This is a distinct advantage, and in fact for almost 
any investigation of current wave shape, the form of circuit shown 
in Fig. 17 is superior to that shown in Fig. 15, since a flexibility of 
control by variation of the mutual inductance is possible; and 
further, the labor of computation necessary for reducing the 
oscillographic data is considerably less by equation (13) than by 
equation (11). Of course care must be exercised to prevent 
extraneous varying magnetic fields from affecting the oscillograph 
circuit, for in such an event, the determination of the line current 
would be vitiated. 

51. Oscillographic Determination of Wave Shape of E.M.F. — 
To secure oscillograms representing wave shapes of voltage by 
means of either the electromagnetic or hot-wire type of instru- 
ment, the current flow must be limited by some form of series 
impedance. If non-inductive resistance is used, the wave shape 
of the current through the vibrator will be identical with that of 
the impressed voltage at the time of recording, though on account 
of the current consumed this voltage wave shape may differ from 
that which would exist on removal of the instrument. If dis- 
tortion of wave shape is desired, the current flow through the 
galvanometer may be limited by inserting series capacity, and at 
the same time decreasing the resistance. If the resistance present 
be small in comparison with the impedance of the condenser, the 
distortion secured by this scheme of connection will be approxi- 
mately proportional to the order of the harmonic. This is ap- 
parent from the fact that the impedance of a condenser changes 
in inverse proportion to change in frequency. 

Very much greater distortion may be secured by the use of a 
circuit of the type shown in Fig. 18. In this diagram, C and r 
constitute the chief current limiting impedance, while the parallel 
circuits, C, r' and L", r", are for the purpose of securing a selective 
current flow through the galvanometer, and may be of much lower 
impedance than the circuit consisting of C and r. The equation 



DETERMINATION OF WAVE SHAPE OF E.M.F. 77 

necessary for the determination of the line voltage E from the 
current I' as represented by the oscillogram is 



E n = [Z n + Z n ' + " jj \l n ' 



(14) 



where Z„, Z„', and Z n " are the impedances for the nth harmonic 
of the respective portions of the circuit. 



r" 



L" 



^AAA-^nnfFn 




L-AAA/^ 



Fig. 18. — Distortion Circuit for Determining Voltage Wave Shapes. 
The current consumed by the entire circuit arrangement is 

I n = (l+^pjl n '. (15) 

To illustrate the use of such a circuit the following numerical 
example is considered. In this case the vibrator current wave 
shape is an assumed one, the values of the various frequency 
components having been selected at random, but with care that 
the total vibrator current be of the proper magnitude. Even 
harmonics are included. 



78 



THE USE OF THE OSCILLOGRAPH 



The equation for the vibrator current i' is 

i' = 0.0158 cos - 0.0054 sin 
-0.0060 cos 2 + 0.0058 sin 2 
-0.0101 cos 3 + 0.0054 sin 3 
+0.0014 cos 4 + 0.0015 sin 4 
- 0.0026 cos 5 - 0.01 12 sin 5 
+0.0004 cos 6 - 0.0064 sin 6 
+0.0079 cos 7 + 0.0022 sin 7 0, 



(16) 



and the wave shape represented thereby is shown by the upper 
curve of Fig. 19. 

Let the circuit constants shown in Fig. 18 have the following 
numerical values. 



r = 200.0 ohms. 
r' = 12.0 ohms, 
r" = 10.0 ohms. 



C = 0.50 X 10-* farads. 
C" = 9.0 X 10" 6 farads. 
L" = 0.150 henrys. 



Table X shows the principal numerical values entering into the 
transformation from the observed harmonics in the oscillogram, 
, that is in i', to those of voltage e. 



TABLE X 

Determination of Voltage Wave Shape from Oscillograph 

Vibrator Current. Connections as per Fig. 18 

(Frequency of fundamental, 60 cycles) 



n 


1 


2 


3 


4 


5 


6 


7 


Z n 


200 
-J5305 


200 
-J2652 


200 
-j 1768 


200 
-j 1326 


200 
-/ 1061 


200 
-^884 


200 
-J758 


Z n ' 


12.0 
-j 294.7 


12.0 
-j" 147.4 


12.0 
-j'98.2 


12.0 
-j 73.7 


12.0 
-.7*58.9 


12.0 
-.7*49.1 


12.0 
-J42.1 


Z n " 


10.0 
+;'56.5 


10.0 
+?113 


10.0 
+7*169 


10.0 
+7*226 


10.0 
+7*282 


10.0 
+7*339 


10.0 
+7*396 


En 
In 


-6620 
+j 20,900 


-630 
+7 574 


-87 
-j 876 


+58 
-j 986 


+118 
-.7*912 


+ 148 
-.7*814 


+ 166 
-j 727 


In 


0.0158 
+7 0.0054 


-0.0060 
-jO.0058 


-0.0101 
-.70.0054 


0.0014 
-.7*0.0015 


-0.0026 
+7*0.0112 


0.0004 
+/ 0.0064 


0.0079 
-./0.0022 


En 


-218 
+/294 


7.11 
+7*0.22 


-3.86 
+7*9.32 


-1.40 
-j 1.47 


9.91 
+7*3.69 


5.28 
+7*0.62 


-0.29 
-j6.il 



(Slide rule accuracy in computations.) 



DETERMINATION OF WAVE SHAPE OF E.M.F. 79 




Fig. 19. — The Distortion Produced by the Circuit of Fig. 18. Upper: Oscil- 
lograph Vibrator Current, i'. Lower: Main Line Voltage, e, by Analysis. 



80 THE USE OF THE OSCILLOGRAPH 

The equation for instantaneous values of e is then, from Table X, 

e = -218 cosd- 294 sine 
+7.11 cos 20- 0.22 sin 20 
- 3.86 cos 3 6 - 9.32 sin 3 6 
- 1.40 cos 4 6 + 1.47 sin 4 6 
+9.91 cos 5 6 - 3.69 sin 5 6 
+5.28 cos 6 6 - 0.62 sin 6 6 
-O.29cos70 + 6.11sin7 0. (17) 

The lower curve in Fig. 19 was plotted point by point from 
equation (17), and serves very well to illustrate the magnifying 
effect of the circuit used on the higher harmonics. It should be 
noticed that for the second and third harmonics, the distortion is 
particularly great on account of the approximation of these 
frequencies to the resonant frequency of the circuits Z' and Z". 

This is indicated by the low values of the ratio y in Table X. All 

of the different frequency components of higher order than the 
first are magnified to about thirty times their normal values in the 
oscillogram. Using a smaller condenser capacity C", or a smaller 
self-inductance L", would prevent the great relative magnification 
of the second and third harmonics, but would still magnify those 
of higher order to a sufficient extent. 



CHAPTER V 

EQUATIONS OF MOTION OF THE GALVANOMETER 

MIRROR IN THE ELECTROMAGNETIC TYPE 

OF OSCILLOGRAPH 

52. General. — In this chapter the equations of motion of a 
mirror, attached to the two supporting strips which constitute the 
galvanometer coil of the electromagnetic type of oscillograph are 
developed, with a view to showing the extent to which the indica- 
tions of such a device are in error when used to record the wave 
forms of uniformly alternating currents. The discussion also 
affords a good illustration of the use of the complex quantity 
notation as applied to alternating quantities other than currents 
and electromotive forces. 

In using the oscillograph to record continuously alternating 
currents, there is probability of two kinds of error, viz., change in 
calibration constant of the vibrator for different frequencies, and 
a time-phase displacement of the mirror deflection from the current 
flowing in the supporting strips. 

Departure of the damping force acting upon the mirror and 
strip from strict proportionality with the velocity of these parts 
would cause the mirror motion to be other than a pure sine wave 
even though the deflecting force be such. Because of the low 
velocity of these two parts, however, it is improbable that such 
an effect could be detected. In calculations relative to the mirror 
motion, the damping force will be assumed proportional to the 
velocity. 

If the moving system is not damped at all, the deflections of the 
mirror will at all times be in phase with the deflecting force for 
frequencies below the natural or free frequency of vibration, while 
for frequencies above this value, the displacement will reverse in 
phase. On account of the damping, the deflections will not be in 
exact phase coincidence with the force acting, but will, for fre- 
quencies below the free vibration frequency, be lagging with 
respect thereto. 

81 



82 EQUATIONS OF MOTION 

The inertia of the moving parts tends to cause a change in the 
calibration constant of the vibrator — particularly for frequencies 
approaching the natural frequency. 

At very high frequencies it is conceivable that the motion of 
the vibrator strips in the intense magnetic field produces an 
induced electromotive force of a magnitude comparable with that 
impressed upon the vibrator circuit. This matter will be referred 
to again. 

53. Fundamental Equations of Motion of the Vibrator Strips. 

— Assume the entire length of the vibrator strip to be acted upon 

by a force 

a cos ut 

per unit of length, by virtue of a current 

i cos (A 
flowing in the strip. Let 

T = tension in the strip, in dynes, and 

m = mass of the strip per unit length, in grams. 

The fundamental differential equation of motion when the strip 
is immersed in a viscous liquid is 

d^s ds S^s 

T te>- k M + aC0S03t = m W (1) 

where' x is measured along the equilibrium position of the strip, 
and s is measured perpendicularly thereto, and thus represents 
the displacement, k represents the damping force per unit length 
of strip when the strip is moving with unit velocity. 

The solution of the above partial differential equation will yield 
a combination of decreasing exponential functions (decreasing 
with time) and trigonometric functions, so that under steady con- 
ditions after the transient terms have become zero the motion of 
the strip at any point in its length will be represented by the 
trigonometric terms which are not affected by the decreasing 
exponentials. Thus 

s = A cos ut + B sin ut, (2) 

where the integration constants A and B are functions of x and 
the physical properties of the system. 

Since s is a simple harmonically varying function it may be 



DETERMINATION OF INTEGRATION CONSTANTS 



83 



represented by the conventional complex quantity notation, and 
the variable t thus eliminated. 
Thus, for any arbitrarily selected origin of time, 



F = deflecting force = /i + .7/2, 

S = displacement of strip at any point = Si + j«2, 

V = -jt = jcoS = velocity, 

A = —rr = -j77 = — oPS = acceleration. 
at at 2 



(3) 



Equation (1) becomes for the permanent condition of operation 



T~-kV + F = mA, 
T 6 ^- jkwS + F = -mco 2 ^. 



Combining terms and placing 

„ _ mat 2 — jku) 



(4) 



T 



N = 



F 
T' 



equation (4) reduces to 






■N, 



the solution for which is 



S =Aic** x + A 2 e-i ax — 



N 



(5) 



(6) 



(7) 



in which Ai and A 2 are arbitrary integration constants which must 
be determined from known boundary or initial conditions. Since 
a 2 is itself complex, the quantity ja is complex, and therefore the 
exponentials become combinations of trigonometric functions and 
exponentials with real exponents. 

54. Determination of Integration Constants. Their De- 
pendence upon the Moment of Inertia of the Mirror. — Let 



2 1 = total length of vibrator strip, or 
I = distance from support at end to mirror at 
the mid-point of the strips. 



(8) 



84 



EQUATIONS OF MOTION 



At x = l f the slope of the strip at any instant must be such as to 
give to the point of contact with the mirror the same acceleration 
as (7) would yield for x — I; this on the assumption that the 
mirror causes no damping over and above that due to the vibrator 
strips, and further that the length of the mirror along the strip is 
negligible as well as any forces due to flexure of the strip at the 
point of contact. These conditions are not realized exactly, but 
to take them into consideration would hopelessly complicate the 
problem. It is not to be expected that the neglect of these matters 
will vitiate the solution to any great extent. 
Let 



4 7 = polar moment of inertia of the mirror about 
a gravity axis parallel to the strips. 

2 d = spacing of the two supporting strips, center 
to center. 



(9) 



The normal acceleration at the point of contact is d times the 
angular acceleration of the mirror, and the torque exerted by each 
half-strip is 



-Td d /\ 
dx} x= i 



The algebraic sign used in this expression for torque applies to the 
first half-strip only, but as all of the four half-strips exert equal 
accelerating forces upon the mirror, each may be thought of as 
accelerating one-fourth of the mirror, and our investigation 
confined thereby to one half-strip only. 



d?s 
dt 2 



I Td? ds\ 

\ z= i I dx\ z - 



(10) 



d 2 s 



In the complex quantity notation,-^ = — &>*$. 



Let 



Td? 
H = —j- , and then at x = l f 

-tfS = - H — • — = S~- 
dx ' dx H 



(11) 



Equation (11) serves as one boundary condition for the deter- 
mination of the integration constants, in that it expresses the 



THE MIRROR MOTION 



85 



value of the first derivative of the function in terms of the function 
itself, at a given value of x. 

As a second condition, since there are two constants to be 
determined, 

S = at x = 0. 
Thus 



N 
A, + A 2 - - 2 = 0. 
or 

Substituting (7) in (11), and then placing x = I, 

jj Ai<t** + jj A*T** - ^ = jaAie** 1 - jaA*-** 1 , 



(12) 



or 



and 



At + A 2 

Solving the two equations of (13) for A\ and A 2 , 
co 2 /co 2 , . \ . . 






(13) 



Ai» 



A 2 = 



a 2 /co 2 . \ . . /co 2 , . \ . . 

N (| 2 - ia )^-| 2 

a 2 /co 2 . \ . . /co 2 , . \ ' 

\H ~ Ja ) ~\H + Ja ) 



(14) 



65. The Mirror Motion. — If these values of A\ and A 2 be 
substituted in (7), the equation of motion for the strip at any 
point will be obtained. Since it is the motion of the mirror only 
that is of interest at present, the introduction of (14) in (7) and 
the substitution of I for x in the resulting equation will give the 
desired solution. S, as given by (7), used with the subscript m 
designates the mirror motion so obtained. 



9 -E 

or 



_ ( e *w _ e -*i) _ 2 jo 
ti 



- 1. 



_ ( e *w _ (-M) — j a (ei al + e - ^) 
ti 



(15) 



80 



EQUATIONS OF MOTION 



Writing (15) in a trigonometric form, that is, substituting sines 
and cosines for the imaginary exponentials, 



S m = 



X 



77 sin al — a 
H 



II 



sin al — a cos al 



which after simplification becomes 



S m — — 
a 



cos al — 1 



// 



sin al — a cos al 



(16) 



which may be taken as the final form. 

Since a 2 is complex, a is also complex, and may be written 

a = u + jv. 

The square root is most easily formed by expressing a 2 as a scalar 
with its angle, and then taking for the scalar value of a the square 
root of the scalar value of a 2 , and for the angle of a, one-half the 
angle of a 2 . Thus, if 

a 2 = a+jb = c/2ji, 
where c 2 = a 2 + & 2 , 

and 



tan 2 = - , 
a 



we have 



a = Vc/fS — Vc cos /3 + j v'c sin j8. 



The trigonometric functions of the complex, a, are 

sin (w + jv) = cosh v sin u + j sinh v cos u, 
cos (u + jy) = cosh v cos u — j sinh v sin w. 



(17) 



Further reduction of (16) by introducing (17) is not desirable, 
except numerically, since the resulting expressions become too 
involved. 

56. Application of Analysis to a Particular Case. — To 
illustrate the application of the foregoing development and to 
show the magnitude of the errors which may be expected in 
practice, we will consider the motion of the mirror in a vibrator 
used by the General Electric Company in their commercial form 



APPLICATION OF ANALYSIS TO A PARTICULAR CASE 87 

of oscillograph. The dimensions of the moving parts are, approxi- 
mately: 

Silver alloy strips, 0.0070" X 0.00075". 
Total length of strips, 0.438". 
Tension per strip, 3.0 oz. 
Spacing of strips, 0.012". 

Glass mirror (thickness increased by 25 per cent to allow for 
cement used in attaching to strips), 

0.060" X 0.017" X 0.0075" thick. 

Using 10.0 as the specific weight for the silver strips, and 3.0 
for the glass mirror, the following values in C.G.S. units are ob- 
tained for the various constants. 

I = 0.555 cm. 
m = 0.340 X 10 -3 grams per cm. length. 
T = 83 ; 500 dynes. 
I = 1.75 X 10~ 8 gram-cm 2 . 
d = 0.01525 cm. 

The quantity k is not known, and its calculation from the known 
dimensions of the system and the viscosity of the liquid would 
be very difficult, but if the assumption is made that the vibrator 
strips are critically damped for the fundamental frequency of free 
vibration, its value may be determined. 

In Byerly's "Fourier's Series and Spherical Harmonics," the 
motion of a string of finite length vibrating in a resisting medium 
is discussed, and in his solution (on pages 113-115) we find, using 
the notation adopted here, the angular velocity of free vibration 
of fundamental frequency to be 

1,/TV 2 fc 2 . 

so that the free vibration frequency when undamped would be 



the familiar equation for vibration of such a type. 
For critical damping, by (18), 



(20) 



88 



EQUATIONS OF MOTION 



For the numerical case in hand, ki = 30.16 and / = 7050. 
These values are, of course, not exact, since the mass of the mirror 
has been neglected, but they serve as a basis for an estimate as to 
the value of k. The free frequency of vibration as given by the 
manufacturers' data is 5000. 

Using the above tabulated numerical constants, three separate 
calculations for various frequencies have been made, taking three 
different values for A;, viz., 

fc = 0, 

k = 0.75 h = 22.62, 

k = 1.50 fci = 45.24, 

the results of which are given in the following table. The quan- 

tity -rp, as given in the table is proportional to the deflection of 

the mirror when unit current is flowing in the strips, and thus 
represents the calibration constant of the galvanometer. The 
angle <f> is the lag in time-phase position of the mirror deflection 
behind that of the current. 



TABLE XI 
Motion op Oscillograph Vibrator 





i = 


* = 0.75t, 


*= 1.50*, 


/ 


s m 


<t> 


s« 


<t> 


s n 


4> 




N 


deg. 


N 


deg. 


N 


deg. 





0.1540 





0.1540 





0.1540 





100 


0.1541 





0.1540 


1.24 


0.1539 


2.48 


500 


0.1556 





0.1547 


6.29 


0.1521 


12.45 


1000 


0.1605 





0.1567 


12.78 


0.1463 


24.53 


1500 


0.1693 





0.1599 


19.66 


0.1389 


35.80 


2000 


0.1834 





0.1641 


27.16 


0.1297 


46.27 


2500 


0.2050 





0.1687 


35.45 


0.1201 


55.82 



Figure 20 shows the curves plotted from the above data. 

The results of this numerical analysis are very interesting, and 
substantiate the statements made in the opening of this discus- 
sion, section 52. In no case can the oscillograph depict the true 
wave shape of a complex wave of current in the vibrator strips. 
If the damping is zero, the deflections are in their proper time- 



APPLICATION OF ANALYSIS TO A PARTICULAR CASE 89 

phase position, but the calibration constant differs for the different 

frequencies — the higher frequency components being magnified 

in the oscillogram. For a damping a little greater than 0.75 hi, 

S 
the calibration constant of the vibrator, -—, will be practically 

unvarying with change of frequency, but in any case with damping 



























N 1 v 








I. 
II. 


Zero Ijamping, 1 

0.75 Critical Damping. 










0.20 


Deg 








III. 


1.! 


OCr 


itica 


IDa 


mpi 


i?. 










0.10 
























































1/ 








0.18 


64 






























































0.17 


56 




















































U, 










0.16 


48 






S»i 


















^111 






















( 












0.15 


40 






























































0.14 


32 




















III 






'n 




































0.13 


24 




























































s 


0.12 


16 








































f 






















0.11 


8 






























































0.10 






400 



800 1200 1600 
Erequency, /. 



2000 2400 



Fig. 20. — Characteristics of Oscillograph Vibrator Motion. Calibration 
Constant and Angle of Lag from Equation (16). 



present, an error in time-phase position occurs. The use of dis- 
torting circuits for the purpose of accentuating the higher har- 
monics in no way increases the percentage error introduced by 
the oscillograph, for such error is dependent upon the frequency 
only, and not upon the magnitude of the deflection (of course 
within the limits of permissible vibrator current). 



90 EQUATIONS OF MOTION 

67. The Intensity of the Magnetic Field Surrounding the 

Strips. — From the observed calibration constant of the instru- 
ct 
ment and the computed value of -^ it is possible to determine the 

flux density of the magnetic field in which the vibrator is placed. 
Let 

L = distance from mirror to observing screen. 
c = observed deflection of light spot per absolute unit of 

current in the galvanometer strips. 
/ = current in vibrator strips, abamperes. 
Then 

D = cl = deflection of light spot on screen. (21) 

For the small deflections used, 

y-j = yj = -r 5 = angular rotation of mirror. (22) 

From equation (22), 

2L at 

F 

By equation (5), however, iV = 7=, so that in (23) 

F = force per unit length = ^- = BI, (24) 

where B = flux density surrounding the vibrator strips. 
Thus 

-!?©■ » 

For the vibrator in question, the calibration constant was 
given as 

0.0060 amperes per mm. deflection, 
from which 

c = 167 cm. per abampere of current. 
Also, 

L = 46 cm., approximately. 

o 

At zero frequency, from Table XI, the numerical value of -r? 
is 0.1540. Substituting these numerical values in equation (25), 
B = 15,010 lines per sq. cm., approximately. 



ELECTROMOTIVE FORCE INDUCED IN VIBRATOR 91 

58. Electromotive Force Induced in Vibrator Strips. — The 
area swept over by a vibrator strip during a half-cycle may be 
determined approximately by assuming the strips to coincide with 
straight lines at the instant of maximum mirror deflection. From 
the known value of flux density, the total change of flux within 
the loop may be determined, and thus the induced e.m.f. 

Suppose the vibrator to be carrying a current of 0.060 amp. and 
thus giving a deflection of 1.0 cm. on the screen. By equation 

(22), then, S m = =-=r = 0.000166 cm., and the total area enclosed 
Z Li 

by the two strips when each is displaced by this amount at the 

center is 0.000184 sq. cm. The flux enclosed by this loop is 

3W. = 0.000184 B = 2.76 lines. 

The effective value of induced e.m.f. is 4.44 $ max . / 10~ 8 volts. 

E = 12.3/ 10" 8 volts. 

Even for / = 5000 cycles per second, this gives only 0.00061 
volts — an amount scarcely comparable with the minimum voltage 
impressed on the vibrator circuit. The resistance of the vibrator 
circuit may be reduced to approximately one ohm, thus requiring 
an e.m.f. of 0.060 volts to produce the amount of current under 
consideration. Thus under the most unfavorable conditions — 
highest frequency and lowest vibrator circuit resistance — the 
voltage induced in the vibrator strips by virtue of their motion in 
the magnetic field is only one per cent of the impressed, so that 
in no one case need any appreciable error be expected to arise from 
the cause here considered. 

69. The Free Vibration Frequency of the System when the 
Inertia of the Mirror is Considered. — The vibrator mirror may 
oscillate freely at an infinite number of frequencies if the damping 
be not too great, and if the damping constant be zero these fre- 
quencies may be determined from equation (16) by equating the 
denominator in the parentheses to zero and then finding by trial 
the values of a> which satisfy the equation. This procedure 
amounts to finding the values of oo for which S m will have a finite 
value even though there be no current flowing in the vibrator 
strips — that is, N = 0. 

Thus, for free oscillation, undamped, 

jj sin al — a cos al = 0. (26) 



92 EQUATIONS OF MOTION 

For k = 0, by equation (5), 

, mo) 2 . fm 

« = -f~> or a = w Vr' 

so that by replacing H by its equivalent, equation (11), 

^sinV^-v/^cosyffco-O, 

from which, by dividing, 

4 /m 7 VmTd? 

tan Vr^ = -^— ' 



tan" 



«/ 



(27) 



(28) 



either of which expressions may be used to determine w. 

A graphical method applied to the first of equations (28) yields 
sufficiently accurate results with very little labor. The two 
curves, 

y' = tany pz« 
and 

V = — f — » 

Id) 

may be plotted, using co as independent variable, and their inter- 
sections noted. The values of w at these intersections are those 
corresponding to free oscillations of the mirror. 

Since the second curve, y", is asymptotic to the axis of «, the 
intersections of the two curves give values of w more and more 
nearly equal to those for which 



tan 



J%U-0, or --«rv/^' 



where n is any integer. 

If no mirror be present the quantity I becomes zero in the 
equations, and then (28) reduces to 



or 



o)\ = tV/ — tan-^oo), 
_|/=o l ▼ m 



(29) 



THE FREE VIBRATION FREQUENCY OF THE SYSTEM 93 

where n is any integer. For n = 1, the fundamental frequency 
of vibration for the strips alone becomes 

'L-£L-n^' (30) 

a result identical with that given in equation (19). 



5 






















































































4 








V 














































































3 
















r 






































































2 
























































































a).. 


1 
















(O 














y" 





















































































6 


)x 1 


o- 1 


















I 




1 




8 




a 




i | 




5 




G 




7 




9-*" 




B 




10 


-1 




































































v/ 


















-2 























































































Fig. 21. — Determination of the Free Frequencies of Vibration of Oscillograph 
Mirror, wi = 32,300. « 2 - 102,500. 

For the vibrator and mirror under consideration, equation (28) 
gives the numerical result, 

70,808 



tan (0.000035415 a) = 



(3D 



Fig. 21 shows the two curves, 

70 808 
y' = tan (0.000035415 co) and y" = — J— , 

from the intersections of which the first two free vibration fre- 
quencies are 

3|300 102,500 _ 

Ztt ait 



94 EQUATIONS OF MOTION 

The first of these values, 5150, may be compared with the free 
vibration frequency for the strips alone, 7050 cycles per second, as 
given by equation (19). The addition of the mirror to the system 
reduces the free vibration frequency by 1900 cycles per second. 

It is to be noticed that equation (29) gives no free vibration 
frequency which is an even multiple of the fundamental frequency, 
obtained by placing n = 1, even though it is known that a stretched 
string or strip may oscillate at any frequency which is a multiple 
of the fundamental. It is to be remembered in this connection that 
equation (16), on which (29) is based, refers to the motion of the 
mid-point of the stretched strip only, and that for even multiples 
of the fundamental frequency this point would be at a node and 
therefore have no motion. For all odd multiples the mid-point 
lies at an anti-node, and thus has a motion — the result shown 
by (29). 



CHAPTER VI 

ALTERNATING CURRENT PHENOMENA IN AN 
IDEAL LINE 

60. The Ideal Line. — By an ideal line, from the point of view 
taken in the preparation of this chapter, is meant one which is 
characterized by four electrical properties per unit of length, viz., 
line resistance, inductance, conductance, and electrostatic capac- 
ity. The line is supposed to be uniform throughout, so the values 
of these four constants or properties are constant for every unit 
of length. These constants must be regarded as the effective 
values of such, since the apparent properties of a transmission 
system carrying alternating currents are dependent upon the fre- 
quency of the line voltages and currents. A discussion of such 
changes will follow, for here it is sufficient to consider the constants 
as effective values and to develop the equations for the alternating 
phenomena in terms of them. 

61. Fundamental Circuit Constants. — In general, for sym- 
metrically arranged polyphase systems it is desirable to take the 
properties of each conductor as referred to the neutral of the 
system, although in the case of single phase transmission with two 
conductors it is satisfactory to consider the conductor with its 
return in determining the physical constants. In the first case, 
voltages are counted for each conductor to neutral, while in the 
second case, the voltage between wires is used. 

Let 

r = line resistance per unit of length, ohms. 
g = line conductance per unit of length, mhos. 
L = self-inductance per unit of length, henrys. 
C = electrostatic capacity per unit of length, farads. 

Since the numerical values of the line constants do not change 
(on the assumption of an ideal line) for a fixed frequency, a sine 
wave of voltage impressed on the line will produce only sine waves 
of voltage and current throughout as long as the load is not of such 

95 



96 ALTERNATING CURRENT PHENOMENA 

a character as to cause a distortion of wave shape. This is in 
accordance with the discussion in section 48, Chapter IV, and 
permits of a representation of the line constants at this fixed 
frequency by means of the conventional complex quantity nota- 
tion, impedance and admittance. 

Z = r -\-j2 vfL = impedance per unit length. 
Y = g + j 2 x/C = admittance per unit length. 



x = 2vfL, z=Vj* + 
6 = 2ir/C, y=V^T6" 2 



(1) 



In the two quantities, Z = r -\- jx and Y = g + jb, it is to be 
noted that the "j" terms are both of the same algebraic sign — 
a matter over which students very often become confused, since 
an admittance, Y, as the reciprocal of an impedance, Z, reverses 
the algebraic sign of the "j, " or imaginary, term. In this instance, 
however, the admittance Y is not the reciprocal of, nor in any way 
related to, the impedance Z — the former being the conductance 
and capacity admittance from wire to neutral, or between wires, 
while the latter is the inductive impedance of the wire, or wires. 

62. Fundamental Differential Equations and Solutions. — In 
the alternating current system the electrostatic capacity is always 
present, and therefore the admittance Y, so that in no case can it 
be said that the current is of uniform value throughout the line 
length — a condition approximately realized in the case of direct 
current transmission over a line with good insulation. For this 
reason the exact solution for the alternating current problem will 
be given first, and approximate solutions discussed afterwards, with 
the rigorous expressions on hand for comparison. 

The fundamental differential equations for current and voltage 
along the line are set up in exactly the same way as for the direct 
current problem, equation (1), Chapter II, using Z and Y in place 
of r and g, respectively, 





f-™;f-*x, 


from which 






drl 

§-ZYI = 0, 




^-ZYE=0. 



(2) 



AUXILIARY CIRCUIT CONSTANTS 97 

Equation (2) is exactly similar to equation (3), Chapter II, the 
constant ZY taking the place of rg (to which ZY reduces for zero 
frequency), so that the complete solution may be written immedi- 
ately, following equation (9), Chapter II. 



e = \ [{e + /„ y/f) t *ra + (e - h y/f) •-*** ] , 



(3) 



where E and 7o are the voltage and current respectively at the 
load and E and I are the voltage and current respectively at a 
distance I from the load end. All of these four quantities are 
in general complex, since they represent electromotive forces or 
currents which are alternating. 

63. Auxiliary Circuit Constants. — Since both Z and Y are 

complex, the two quantities \ y and VZY are, in general, both 

complex. Let 

V=VZY = a+jl3 (4) 

serve as a defining equation for V, a, and /3, which thereby become 
constants per unit of length of the transmission system, a and /3 
may be explicitly expressed in terms of the fundamental line con- 
stants by the following method : 

(a+i/3) 2 = (r+jx)(g+jb), 
or 

a 2 + 2 ja& — /3 2 = rg — xb + jgx + jrb, 

from which by equating reals and imaginaries, 
a 2 — /3 2 = rg — xb, 



(4a) 
2 aP = gx -f- rb. J 

From these two equations, as Steinmetz shows (Transient Electric 
Phenomena and Oscillations), 

« - V| {zy -xb + rg),) . g . 

= Vi (21/ + xb - rg)\ 

Though convenient for some purposes it is not desirable to use 
equation (5) for the calculation of numerical values of a and /3, 
Darticularly when the line resistance r and leakage g are small in 



98 ALTERNATING CURRENT PHENOMENA 

comparison with x and b, respectively. In such an event the 
product zy differs very little from xb, and in the expression for a 
the resulting difference, zy — xb } would be difficult to obtain with 
accuracy. Besides, in any case the numerical work required is 
excessive. It is better to form 

Z m r+jx = z/0f 

Y = g+jb = yld JI 

VZY m « + j0 - V - Vzy /h (0. + B y ) = vl$, 

a = v cos 6 V , 

= v sin 8 V , 

and at the same time to form 

For convenience in writing, let 

U = y | = u/Ou. (6) 

64. Complete Expansion of Exact Solution. — If. now, the 
substitution 

e ±w = e ±w+#o m € ±<*i ( cos ft ± j sin ^) ( 7 ) 

be made in equation (3), we obtain 

E = \ (E + Uh) (? l (cos 01 + j sin 01), 

+ H^o - tf/o) c- a/ (cos 0Z - ; sin #) 

/ = ^(l o -\-jjE^ l (co8 0l+jsm0l) 

+ | (l - jj E)j e-^fcos 01 - j 'sin 0Z) . (8) 

Equation (8) is particularly well adapted to the physical inter- 
pretation of the mathematical expressions, although, as in the 
solution of the problem for the transmission of direct currents over 
leaky lines, a more convenient form for computing purposes may 
be used. 

65. Interpretation of the Various Terms in the Expanded 
Solution. — In equation (8) consider first the quantity 

cos 01 ± j sin 01 = «*** 



INTERPRETATION OF VARIOUS TERMS IN SOLUTION 99 

The absolute value of this expression, as the square root of the 
sum of the squares of the two components, is alway unity, so it is 
evident that the only function such a factor can perform is to 
rotate, or change the phase position of, a vector representing an 
alternating quantity. In fact this quantity is called a "rotating 
operator," because it rotates any vector which is affected by it as 
a multiplier, through an angle pi. 

cos pi + j sin pi = e+#" 

rotates a vector in a counter-clockwise direction, and 
cos pi — j sin pi = e~#' 

rotates a vector in a clockwise direction — in both cases by an 
angle numerically equal to pi radians — while the length of the 
vector remains unchanged. 

Returning to equation (8) it is seen that the line voltage (the 
same remarks apply to the line current) consists of the vector sum 
of two apparently distinct components. For increasing values of I, 
that is, going from load towards generator, the vector representing 
the first component, 

E' - | (#„ + UI ) e l (cos pl + j sin pi), (9) 

rotates in a counter-clockwise direction by an amount proportional 
to Z, and at the same time increases in length on account of the 
factor e"*. Counter-clockwise rotation of a vector signifies an 
advance in phase position of the harmonically varying quantity 
represented conventionally by this vector. Since the variation 
of e' with time is simple harmonic at a fixed point in the line, and 
for different points along the fine the phase position of e' advances 
proportionally to the distance, it appears that this quantity E' 
is merely the vector representation of a voltage wave moving in 
a negative direction along the line (generator towards load) and 
decreasing in magnitude as it moves along. Since E' at points 
near the source of power is advanced in phase position with respect 
to E' at points more remote, it is obvious that the direction of the 
wave motion of this component is opposite to our arbitrarily 
selected positive direction along the line. For increase in Z, this 
component increases, but speaking of the wave represented by 
E', it may be said to decrease in magnitude in the direction of its 
propagation. 



100 ALTERNATING CURRENT PHENOMENA 

As to the remaining component, 

E" - | (E - l//o) e-" 1 (cos /M - j sin 01), (10) 

it is seen that the vector which this represents rotates in a clock- 
wise direction as I increases, and at the same time decreases in 
magnitude according to the multiplier e - "*. E" is thus the 
vector representation of a wave moving along the line from the 
load towards the generator — decreasing in value as it goes. 
According to the above physical interpretation of the meaning of 
the two terms in the expression for the line voltage, the e.m.f. at 
any point is made up of the sum of two separate waves moving 
in opposite directions; a main wave, £", moving from generator 
towards load, and a reflected wave, E" , moving from the load 
towards the generator. Both of these waves decrease in magni- 
tude at the same proportionate rate, in the direction of their 
propagation. This decrease in amplitude of each of the separate 
waves is due to the loss of energy in the resistance and leakage of 
the line, that is, the vhr and e 2 g losses. For a line with no resistance 
or leakage the waves do not change at all in magnitude, though 
the effective line voltage or current of course varies along the line 
on account of the different phase positions at which the main and 
reflected waves combine into the resultant. 

As stated before, the current equation may be interpreted in 
exactly the same manner as has been done with the expression 
for v6ltage. The two components of the resultant current are 

r = l( h + b Eo ) ** 1 (cos fil + j sin fil)t (10a) 

which represents the main wave, or wave traveling from generator 
towards receiver, and 

1" = l(lo-jj Eo) r* (cos 01 - j sin 01), (10b) 

which represents the reflected wave, traveling in the opposite 
direction from the main wave. 

66. Example of Calculation of Fundamental Constants in the 
Equations. — Before going further with a discussion of the 
equations, a numerical example will be considered in order to 
illustrate the method of determining the numerical values of the 
quantities which enter into the equations and to form a basis 
for the construction of the vector diagrams representing the 
phenomena. 



EXAMPLE OF CALCULATION 



101 



TABLE XII 
Calculation of Auxiliary Constants of Transmission Circuit 
Fundamental line constants, per mile to neutral: 
r = 0.275 = 0.15X10-°, L = 0.00204, C = . 0146 X 10-*, 

/ = frequency = 60 cycles , Log L = 7 . 309630-10, Log C - 2 . 164353-10. 



X = 2ir/L 


log/ 

log 2 7r/ 

log a; 

logr 


1.778151 
2.576331 
9.885961-10 
9.439333-10 






X 

tan 6 Z = - 
r 


log tan 6 Z 
log sin Z 


0.446628 
9.973872-10 






X 


log z 
0* 


9.912089-10 
70° 19' 26.42" 


Z = r +jx 


= Z/Jz 


sin Z 


b = 2tt/C 


log 6 
log? 


4.740684-10 
3.176091-10 






6 

tan 6 y = - 

9 


log tan 6 y 


1.564593 






log sin U 


9.999839-10 






b 

y = — — 

sin y 


logy 
0y 


4.740845-10 
88° 26' 20.16" 


Y = g+jb 


= ylh 


v 2 = zy 


2 log v 


4.652934-10 


V 2 = ZY = 


zy/dz + By 


2 Ov — 6z ~T uy 


2 0„ 

0„ 

log i> 


158° 45' 46.58" 
79° 22' 53.29" 
7.326467-10 


V = a+j0 


= v/e v 


a = v cos 6 V 
= v sin 0„ 


log cos 6 V 

log sin 0„ 

log a 

log/3 

. 360 

log o- 


9.265453-10 
9.992498-10 
6.591920-10 
7.318965-10 

1.758123 






«°_ 360 * 


log/3° 
0° 


9.077088-10 

Q. 000390769 
0.119423 






«*«-* 

y 


2 log u 


5.171244 


tf-Vf- 


■■ u/0 u 


2 0u = 0« — 0y 


2 0„ 

log M 

U 

l0g ^ 


-(18° 06' 53.74") 

2.585622 

- (9° 03' 26.87") 

7.414378-10 


b-fz- 


■;*=* 




U 


385.143/- (9° 03' 


26.87") 






1 
E7 


0.00259644/9° 03' ! 


26.87" 





102 



ALTERNATING CURRENT PHENOMENA 



The numerical calculation of the line constants is given in detail, 
and, as is most convenient, is carried out by means of logarithms. 
Even when carried out with the accuracy afforded by six-place 
logarithm tables the amount of work required is very small, no 
work being required other than that indicated in the table. In 
general it would not be advisable to make such a computation with 
six-place tables, since the accuracy of the computation is very 
much greater than that of the fundamental data. Five places are 
usually sufficient. 

67. Vector Diagrams Representing the Exponential Form of 
Equations. — Suppose that at the load the e.m.f. between one 
wire and neutral is 50,000 volts, and that the load current is 25.0 
amperes. Suppose the power-factor of the load to be such as to 
cause the current to lag 25.0 degrees behind the voltage in time- 
phase position. Using the numerical values of the constants as 
determined from the calculations in Table XII, with equation 




■Fig. 22. — The Vectors Representing the Main, Reflected, and Resultant 
Voltage Waves at the Load End. 

(8) the vector diagrams representing the separate components, as 
well as the resultant, of the voltage and current along the line will 
be constructed. 

If the load voltage be selected as the reference vector (that is, 
the arbitrary origin of time so selected as to make E a cosine 
variation), we have 

E = 50,000/0^ volts, 
7 = 25.0 / — 25° amperes, 
and thus 

Uh = 9628.58 /- (34° 03' 26".87) vo it Sj 

jj E = 129.822 /9° 03' 26".87 amperes. 

Fig. 22 shows the vectors, E , h, Uh, and the two vectors 
representing the component waves at the receiving end, 

2 Eo' = E + Uh and 2 E " = E - Uh. 



VECTOR DIAGRAMS 103 

Fig. 23 shows the analogous current vectors, with E again shown 
as the reference, E , I , •= Eo, and 

2/,' = /„ + ^o and 2/o" = h-jjE*. 

If now a numerical value be assigned to I, say 100 (miles), the 
vector 2 E ' will be rotated in a counter-clockwise direction 




Fig. 23. — The Vectors Representing the Main, Reflected, and Resultant 
Current Waves at the Load End. 

through an angle fil, or in this case 11.9423 degrees, and at the same 
time increased in length by the factor 

e l = e O.039O769 = 1.03985. 

2 E a " rotates by an equal amount in a clockwise direction, and 
decreases in length by the factor 

r* 1 = 0.96168. 

One-half the vector sum of these two vectors is the vector repre- 
senting the actual line voltage at the distance of one hundred miles 
from the load. This resultant is very easily obtained by drawing 
a vector from the origin to the mid-point of the line joining the 
extremities of the component vectors. 

In Fig. 24, E is the load voltage, to scale; 2 E<f and 2 E " are 
twice the load-end values of the main and reflected waves, respec- 
tively. The curve, Locus of 2 E', indicates the path followed by 
the end of the vector in question for continuously increasing values 
of I. For each 400 mile point in the line, the vector is drawn in 
the diagram. The last vector so drawn is for a line 2000 miles in 
length. For I = 2000 the rotation of 2 E' is 238.846 degrees, and 
the factor (? l is 2.1848. The curve, Locus of 2 E", indicates the 
path followed by the end of the vector 2 E" for continuously in- 
creasing line lengths. The spiral is in this case a decreasing one, 

and for I = 2000, the factor r" 1 = 1Q . Q = 0.45770. 

2.1848 



104 



ALTERNATING CURRENT PHENOMENA 



The resultant of the two waves, E' and E" is given by the vector 
E, the end of which moves along the curve, Locus of E. 

The rotation of the vector E is not proportional to the line 
length as is the case with the separate components, since it is 



Locnsof2E' 




Fig. 24. — Diagram of Voltage Vectors for Increasing Line Length. 
Subscripts Refer to Line Length. 



made up of the vector sum of two vectors rotating in opposite 
directions and changing in length as they rotate. 

Fig. 25 contains the current vectors, all of which are treated in 
the way just described for the voltage vectors. For the load 
selected as illustration, the e.m.f. UI , which is added to and 
subtracted from the e.m.f. E in order to form the initial vectors 



VECTOR DIAGRAMS 



105 



for the separate waves, is considerably smaller than E , and there- 
fore the initial vectors E ' and E Q " do not differ very much from 
E . Thus, for distances up to approximately 100 miles from the 
load end, the resultant voltage vector does not change very much. 



Loons of 21 ! 




Fig. 25. — Diagram of Current Vectors for Increasing Line Length. 
Subscripts Refer to Line Length. 



In the case of the vector diagram of currents, the component 
Yj Eq is several times the load current, so that the current at the 
receiving end, 7o, is made up of the sum of two vectors nearly 



106 



ALTERNATING CURRENT PHENOMENA 



equal in length and displaced from each other by approximately 
180 degrees. The rotation in opposite directions of these two 
component vectors produces a resultant vector / which varies 
very rapidly for small increase in line length, as may be noticed in 
Fig. 25 from the manner in which the curve, Locus of /, starts 
from its initial point at 7 . 




2000 



Fig. 26. — Voltage and Current Vectors. Numbers Indicate Line Length. 



The two loci of the ends of the resultant voltage and current 
vectors are reproduced in Fig. 26, the vectors for the end of each 
400 mile section of line being drawn. For any point in the line, 
the power-factor is given by the cosine of the angle between the 
vectors representing E and I for this length. For example, in 
Fig. 26, where the numbering indicates the length of line to which 
each vector belongs, for the end of the 800 mile section the current 



OPERATING CHARACTERISTIC FROM VECTOR DIAGRAMS 107 



is ahead of the voltage in time-phase position by an angle of 
approximately 10.5 degrees, while at a point 1200 miles distant 
from the load the current is lagging by 22.8 degrees. Again, at 
2000 miles, the current leads by 25.8 degrees. For increasing line 
length this power-factor angle oscillates about an angle equal to 

that of the quantity U = y y, as will be shown later. Since the 

load as well as the line losses must be supplied by an average flow 
of power along the line in a negative direction (generator towards 
load) it is evident that this phase angle can never be greater than 
90 degrees, for then the average power transmitted (EI cos <£) 
would reverse in sign, and thereby indicate a flow of energy in the 
opposite direction. In fact, for any line with losses (r or g present) 
the power-factor angle can never equal 90 degrees, for then there 
would be no average power passing the point in question to supply 
the line losses in the portion of the line more distant from the 
generator. 

68. Operating Characteristics from Vector Diagrams. — From 
the original drawings for the preceding figures, in which the vectors 
were inserted for every 200 mile section, the following values were 
obtained: 



TABLE XIII 

Distribution of Electrical Quantities Along a Long Trans- 
mission Line 



Length, 
miles 


E M ., volts 


I M ,, amp. 


Power-factor 
angle, degrees 


K.W. per phase 


Effic, per cent 





60,000 


25.0 


25.0 lag 


1130 


100.0 


200 


48,900 


49.5 


58.5 lead 


1245 


90.9 


400 


40,500 


94.5 


65.0 lead 


1612 


70.1 


600 


28,900 


128.0 


49.8 lead 


2385 


47.3 


800 


24,300 


143.2 


10.5 lead 


3410 


33.2 


1000 


34,700 


139.5 


19.5 lag 


4550 


24.8 


1200 


49,200 


122.8 


22.8 lag 


5550 


20.4 


1400 


60,600 


107.0 


8.5 lag 


6400 


17.7 


1600 


65,700 


112.5 


13.2 lead 


7250. 


15.6 


1800 


65,000 


140.0 


26.2 lead 


8150 


13.9 


2000 


61,700 


173.5 


25.8 lead 


9620 


11.7 



(The above values were determined by measuring the vector diagram, so may be slightly in- 
accurate.) 

The above numerical results are plotted in Fig. 27, with line 
length as independent variable. 



108 



ALTERNATING CURRENT PHENOMENA 



160 



no 



uo 



U> I 



.SI 



oo 



•to 



20 



K.y 


: 




































J Dea. 


70 








































60 






















\i 




















50 


(SO 








































10 










































30 


Ox 










i 


has. 


\a 


igli 
























20 
























E/ 


















10 





















































































-10 


I'M 








































-SO 










































.80 


20 








































-40 


















■^p< 


►we 






















-60 


10 
































































































































K.W. 

x 

io- 1 



0.2 



0.4 



0.6 0.8 1.0 1.2 1.4 1.6 

Line Length, Thousands of Miles 



2.0 



Fig. 27. — Voltage, Current, Power, and Phase Angle in a Long Line. 

69. Limiting Forms of Vector Diagrams for Infinite Length 
Line. — In this figure the curves representing E and I approach 
the simple exponential curve in form when the line length becomes 
infinite. This follows from the disappearance of the term repre- 
senting the reflected wave, in equation (8), because of the de- 
creasing exponential. The equations of the limit-curves are thus 

E = Eo'r l , 
and 

I = /oV, 

in which scalar values are used. The wavy appearance of the 
curves in Fig. 27 for the line under consideration is due to the 
combination of the main wave with the reflected wave, which, for 
the comparatively short length of line, has not disappeared. 

The power curve, on the other hand, does not approach a simple 
exponential curve as a limit, as will be shown under the discussion 
of power relations and line losses. 

70. Instantaneous Values from Vector Diagrams. — If it is 
desired to know the instantaneous distribution of current and 
e.m.f. along the line it is, of course, only necessary to draw the 
circles on the vectors in the diagram as diameters, and then to 



INSTANTANEOUS VALUES FROM VECTOR DIAGRAMS 109 

measure the intercept on the time radius drawn through the origin 
at an angle with the reference vector equal to the product of the 
angular velocity of alternation (co = 2rf) and the elapsed time 
between the zero instant and the instant for which the distribution 
is desired. If the diagram represents effective values, the inter- 
cepts must be multiplied by the square root of two in order to 
secure instantaneous values. 

Kilo- 
volts 











































80 
























e. 


















60 


























/> 


























































40 




















































































20 
























e" 
































Lin 


c r. 


me 


li 














11 D 


idtf 


'(IS ( 


)fJI 


ilea 
















1 


- 




( 


1 


**d 


1 


1 


) 


i 


> 


1 


4 


1 





.1 


S 


1 


1 










e£. 
































-20 


















































e^, 


































-40 










V 








































e, 


































-60 




















































































-80 



Fig. 28. — Instantaneous Distribution of Voltage 0.075 Second After 
Positive Maximum of E . 



By selecting E as the reference vector in this numerical illus- 
tration, we have, as functions of time, 

e = a/2 50,000 cos 2 rft, 

referred to the instant at which e passes through its positive 
maximum, and 

U = ^2 25.0 cos (2tt/Y - 25°). 

Let the curves be plotted showing the instantaneous distribution 
of current and voltage 0.075 second after the origin of time. The 
time radius will have turned through 0.075 X 60 = 4.50 complete 
revolutions, or will be displaced from the reference line by an angle 



110 



ALTERNATING CURRENT PHENOMENA 









































Amp. 




































/' 
















































TOO 




































/ 
















































120 




















































































80 








4" 












































































40 














































I.i 


It' I 


«M 


-tli 












III 


ndiBda 


of Mila 


1 
















1 


1 






S 


1 j 


' 


1 




1 


1 


2 


1 


I 


16 


18 


20 




J 
























i" 














-40 


















































































-80 






\. 














































































-120 



Fig. 29. — Instantaneous Distribution of Current 0.075 Second After 
Positive Maximum of E . 









































Watts 


























Vul{ 


at-'t'^ 


'"' 








V 




X 

vr* 


























J 


f 


„-~ 


— 


4j« 


rrer 


t S 


\ 


14 




























y 








N 


v 
v 


\ 
> 


12 
























/ 


/ 












\ 


V 


10 






















/ 


r 




Poa 


er 










\ 


8 




















/ 


7 




















6 


















/ 


>■'/ 

1 






















4 
















• 




< 






















2 














s 




/ 
1 y 


/\. 


lie 


.('11 


.Mil. 


Hu 


uln 


■,1s ( 


<t 1 


Dm 














1 




J^ 


y 




1 

/ 


\ 


I 


1 


1 


1 


1 


4 


1 


; 


i 


- 


•> 


°-2 








**" 








/ 


























-4 














/ 
/ 




























-6 












/ 
/ 








































4 

/ 


/ 






































• 


/ 
t 






































y 







































Fig. 30. — Instantaneous Distribution of Power 0.075 Second After Positive 
Maximum of E a . Voltage and Current Curves Reproduced from Figs. 28 
and 29. 



LOCI OF THE ENDS OF THE COMPONENT VECTORS 111 

of 180 degrees. The square root of two times the intercepts of 
the various circles on this radius gives the values which are plotted 
in Figs. 28, 29, and 30. 

71. Instantaneous Values of Power at Different Points. — In 

Fig. 30, the instantaneous values of power are obtained from the 
products of the corresponding instantaneous values of current and 
e.m.f. It is interesting to note that at this particular instant, for 
the section of line between I = 600 and I = 800 miles, the power 
is negative — that is, the flow of energy is from load towards 
generator — while on either side of this section the energy flow is 
from generator towards load. At any point in the line at which 
the power-factor is not unity, these negative values of instantane- 
ous power may be observed at some instant. From Fig. 27 it is 
seen that at the distances of 40, 870, and 1480 miles from the load 
the angle of lag is zero (power-factor unity), and thus the in- 
stantaneous power at these points cannot be negative at any time. 
These large variations in the distribution of instantaneous power 
along the line are of course occasioned by the continual and periodic 
redistribution of the energy stored in the line self-inductance and 
electrostatic capacity. 

To show the variation in power distribution along the line from 
instant to instant, Fig. 31 contains curves giving the instantaneous 
values of power for six successive intervals of time, beginning with 
the instant at which E passes through its positive maximum 
(t = 0) and covering one-half of a cycle. 

No. 0, t = 0. 

.No. 1, t = 1/720 second, 1/12 cycle. 
No. 2, t = 2/720 second, 2/12 cycle. 
No. 3, t = 3/720 second, 3/12 cycle. 
No. 4, t = 4/720 second, 4/12 cycle. 
No. 5, t = 5/720 second, 5/12 cycle. 

Though this set of curves covers only a half-cycle of voltage or 
current, it is sufficient to cover a whole cycle of the power wave, 
since this latter is of double the frequency of its two components, 
voltage and current. 

72. Equations for the Loci of the Ends of the Component 
Vectors. — The curves traced out by the ends of the vectors 
which represent the component waves are logarithmic, or equi- 
angular, spirals. Take, for example, the locus of the end of the 



112 



ALTERNATING CURRENT PHENOMENA 



vector representing the main wave. Its initial length is eo', and 
for increasing distance from the load end of the line, its length 
is represented by the equation 

e' = eoV, (11) 











































fa 










































20 








































Y 


18 




































X 






16 










































11 






























0/ 












12 


























^\ 
















10 




















V 






















8 












I 




JL 


























6 






•V 




































4 










































2 



















































































-2 










































-4 












Dis 


tuiu 


o f r 


Dill 


Lou 


din 


Hu 


mlr 


eils 


rfl 


files 


. 








-6 







■> 




4 









8 




10 




12 




u 




10 




18 




20 



Fig. 31. — Instantaneous Distribution of Power 0, 1, 2, 3, 4, and 5 Twelfths 
of a Cycle after the Positive Maximum of #0. Numbers on Curves Indicate 
Twelfths of a Cycle. 



and the angle the vector makes with its initial position is 

e = 0Z. 



Eliminating the parameter I by I = 



e' = e»'f\ 



(12) 



(13) 



which is the equation of the exponential curve in polar coordinates. 
Let <f> = the angle between the tangent to the curve and the 
radius vector. Then from the geometry of the figure, 



^ = ecot*. 



(14) 



CONSTRUCTION OF EQUIANGULAR SPIRALS 113 

By differentiating (13), 

de' , a ^e a . ,,_,. 

*-*V =r~ (15) 

so that by comparing equations (14) and (15), there results 

cot« = ~ (16) 

From this it is seen that cot 0, and therefore <j>, is a constant. It 
is from this property that the curve derives the appropriate name 
of "Equiangular Spiral." 
In the expression 

V - VZY = v/dy 

and, as previously shown, 

a = v cos 0„, 
j8 = v sin 0„, 
so that 

- = cot 6 V . 

Thus, from (16), the angle <f> between the tangent to the curve 
and the radius vector is the same as the angle 6 V , which, in turn, is 
equal to the mean of the impedance and admittance angles. 

= 0„ = H0. + 0v). (17) 

For the decreasing spiral, the angle is the same as above — 
the rotation of the radius vector in the opposite direction causing 
a diminution instead of an increase in the length of the radius. 

73. Mechanical Construction of Equiangular Spirals. — The 
equiangular property of these curves suggests the possibility of 
constructing them by some purely mechanical means. This may 
be done by the use of the apparatus shown in Fig. 32, in which a 
small sharp-edged tracing wheel is mounted in a frame so that the 
plane of the wheel may be set at any desired angle <f> with the axis 
of the instrument. Along the axis of the instrument is a narrow 
slit, which slit slides over a pin placed at the center of the vector 
diagram, at 0. A scale of degrees and a vernier may be placed on 
the rotatable mounting of the tracing wheel so that the angle 
may be set off accurately. The zero of the scale may be located by 



114 



ALTERNATING CURRENT PHENOMENA 



ascertaining by trial the position at which the wheel is to be set 
so as to trace out a circle. By pressing the wheel firmly against 
the paper and rotating the entire apparatus about the point 0, an 
equiangular spiral will be traced out as shown — the angle <j> at 
which the instrument is set being the same as the angle <f> in 
equations (16) and (17). As shown, rotation in a counter-clockwise 



#-» B \+(*«+M 




Fig. 32. — Polar Exponential Curve Tracer. 

direction gives the increasing spiral, and rotation in the opposite 
direction, the decreasing spiral. 

Using such a device, it is only necessary to draw in the diagram 
the two initial vectors, 2 E ' and 2 E ' ', and with the angle <f> set off 
on the vernier and scale, to place the tracing wheel upon the end 
of each vector and to draw in the spirals. This done, the vectors, 
2 E' and 2 E", corresponding to any particular line length are 
obtained by drawing the lines which make angles /3Z with the 
initial vectors, from the center of the diagram to the curves. The 
resultant vector is obtained as before, by drawing the line from the 
origin to the mid-point of the line joining the extremities of the two 
component vectors, 2 E' and 2 E" . 

By the use of such a device, comparatively accurate solutions 
for lines of different length, but with constant receiver load, may 
be obtained very rapidly after having at first computed the 
numerical values of the auxiliary constants, a, /3, and U, as per 
Table XII. 

74. Introduction of Hyperbolic Functions of Complex Vari- 
ables. — In the preceding section the manner of the variation 
of the line phenomena with change in line length has been dis- 
cussed. The vector diagrams, for any distance from the receiving 
end of the line have been shown to be very easily obtained by the 
use of a mechanical device for tracing the exponential curves. 



CALCULATION OF CONSTANTS IN EQUATIONS 115 

Such procedures are particularly applicable to cases of constant 
receiver load and variable line length. 

In dealing with a line of given length over which a variable load 
is to be supplied, a more convenient form of expression than the 
exponential in equation (8), as exemplified in the vector diagrams 
just given, may be obtained by using hyperbolic functions. 

By combining terms in equation (3), the complete solution may 
be put into the form 



(18) 



E = E \ (^ ZYl + e^™) + Jo\/f \ (* VZYI ~ e~ VzYl ), 

I = 7o \ (e V ™ + e-^0 + %>SJ\ \ (e V ™ - e" V ™), ( 

which, by introducing 

V=VZY, and U = \Jy, 

and the hyperbolic functions for the combinations of exponentials, 
become 

E = E cosh Vl + hU sinh VI, 



I = h cosh Vl + E jj sinh VI 



(19) 



Naturally, equation (19) is identical in form with equation (14) 
of Chapter II for the direct current system. V = VZY is of 
course complex, so that the hyperbolic functions of a complex 
quantity are required. 

V = a+jfi. 

cosh VI — cosh al cos 01 + j sinh oil sin f$l, | 
sinh VI = sinh al cos fil + j cosh al sin /3Z. J 

Equation (19) for the alternating current problem may be 
carried through all the developments as are given in Chapter II 
for the direct current system, keeping in mind, of course, the fact 
that the complex quantity notation must be retained. 

75. Calculation of Constants in Equations for a Line of Given 
Length. — The utmost convenience and dispatch are to be found 
in the use of equations (19) for the determination of the operating 
characteristics of a transmission system, and particularly if a 
combination of analytical and graphical methods is employed. A 
numerical example will serve best to indicate the method. 




116 



ALTERNATING CURRENT PHENOMENA 



Let it be required to plot curves showing the operating char- 
acteristics of a transmission line whose constants are those given 
in Table XII and whose length is 400 miles, for the condition of 
constant voltage at the load end. 

The first step in the work is to obtain the numerical values of 
the three coefficients 

cosh VI, U sinh VI, and jj sinh VI, 

as required in equation (19). From the values of a and /3 given 
in Table XII and using tables of the logarithms of hyperbolic 
functions, the computations may be conveniently carried out as 
shown below in Table XIV. 



TABLE XIV. 

Computation of Constants in Hyperbolic Expressions for 

Transmission Line Phenomena 

Line length = 400 miles 



From Table XII, a 

a 

al 
f* 

log cosh al 

log sinh al 

log sin pi 

log cos pi 

log A c = log cosh al cos (31 

log B c = log sinh al sin pi 

log tan B c 

log cos B c 

log cosh VI 

log A, = log sinh al cos pi 

log B, = log cosh al sin pi 

log tan B t 

log sin B, 

log sinh VI 

From Table XII, log U 

log U sinh VI 

log jj sinh VI 

cosh VI 
U sinh VI 



Results of 
Computation 



jj sinh VI 



0.000390769 

0.119423 degrees 

0.156308 

47.769 degrees 

0.00528 

9.19575-10 

9.86949-10 

9.82745-10 

9 . 83273-10 cosh VI = A c +jB e 

9.06524-10 

9.23251-10 

9.99376-10 

(9 . 83897-10) 79.693 degrees 

9 . 02320-10 si nh VI = A . + jB. 

9.87477-10 

0.85157 

9.99574-10 

(9 . 87903-10) /81 .989 degrees 

(2 . 58562) /- 9.057 degrees 

(2 . 46465) 772.932 degrees 

(7 . 29341-10) 791.046 degrees 

. 69020 /9.693 degrees 
291.51 /72.932 degree s ohms. 

0.0019652 /91.046 degrees mhos. 



Thus, numerically, 

E = E [0.69020/9^693] + 7 [291.51/72^932] 
I = I [0.69020 /9°.693 ] + E [0.0019652 /91 °.046] 



GRAPHICAL TREATMENT FOR LOAD-END CONDITIONS 117 

76. Graphical Treatment for Various Load-end Conditions. — 
From this point on, the remainder of the solution may be obtained 
graphically by constructing the vector diagrams representing 
equation (19) on the basis of the numerical values furnished by 
Table XIV. Since the load voltage is supposed to be kept 
constant, the vector diagram is constructed on E as reference 



E for P.F. Angle »0 and I -= 100 Amp, 



I U «inh VI 




^ Voltage Scale 

! — I 1 1 1 1 1 1 1 1 1 

10000 20000 

Fig- 33. — Voltage Diagram for Various Load Currents. Based on 
Equation (19). 

vector. Let E — 50,000 volts, effective. Fig. 33 is the voltage 
diagram, representing the equation 

E = E cosh VI + IqU sinh VI. 

The resultant voltage at the generator is made up of two parts — 
one part proportional to the load voltage and the other to the load 
current. The vector representing E cosh VI = 34,510 /9.693 deg. 
is drawn at an angle of 9.693 degrees from E and with a length 
to the same scale as E representing 34,510 vols. This vector, 
which is one of the two which go to make up the resultant vector 
for E, remains fixed regardless of the load current 7 . In the 
diagram, A designates the end of this vector. 
To complete the diagram and make it applicable to any and all 






118 ALTERNATING CURRENT PHENOMENA 

values of load current at any power-factor, select a base value of 
current of, say, 100 amperes at unity power-factor. Then 

h = 10 0/0 deg., 

and IoU sinh VI = 29,151 /72.932 deg. volts. 

The resultant generator voltage at E is obtained by adding the 
two vectors in the ordinary manner, giving, for this particular 
load, E = 54,30 0/38.2 deg. by measurement of the diagram. This 
gives the solution for only one value of load current, but it is readily 
seen that for any other value of load current at this power-factor, 
the solution is obtained by taking the proportional value of the 
vector I U sinh VI, and, of course, of AE. For example, the line 
AE may be divided into five equal parts, giving thereby the values 
of generator voltage for I = 0, 20, 40, 60, 80, and 100 amperes. 
An extension of the vector beyond the length of AE gives solutions 
for load currents greater than 100 amperes. 

The length of the vector I U sinh VI depends only upon the 
absolute value of I , while the angle it makes with the reference 
line OE depends only upon the power-factor of the load, and is, 
in fact, equal to the sum of the power-factor angle and the angle 
of the factor U sinh VI. Thus, for any power-factor of load, it is 
only necessary to turn the vector I U sinh VI, as drawn in the 
diagram for unity power-factor, through an angle equal to the load 
power-factor angle. Rotation of this vector also rotates the line 
AE through an equal angle. The dotted lines in the figure indicate 
the vectors representing the solution for a load current of 100 
amperes lagging 25 deg. behind the load voltage. Arcs of circles 
drawn through the points subdividing the line AE provide solu- 
tions for the intermediate values of current at any power-factor. 

To obtain the value of the generator current we again take the 
sum of two components — one proportional to Eq, and therefore 
constant in this particular case, and the other proportional to 7 . 
Fig. 34 is the diagram of current vectors with E again used as 

reference. The quantity #077 sinh VI required by equation (19) 

is equal to 

(50,000 /0 deg. ) ( 0.0019652 /91. 046 deg. ) 

= 98.26 /91.046 deg. amperes. 



GRAPHICAL TREATMENT FOR LOAD-END CONDITIONS 119 

This current is represented by the vector OA. To it is to be added 
the remaining component I cosh VI which, in this case, depends 
upon the load current and the power-factor. Select a base value 



I for P.F.Angle«0 and I ~100 Amp. 



B^sinhVlefZZ 




Fig. 34. — Current Diagram for Various Load Currents. Based on 
Equation (19). 

of current of one hundred amperes at unity power-factor as was 
done in constructing the voltage diagram. Then 

7o cosh VI = (100/Odeg.) (0.69020 /9.693 deg. ) 
= 69.0 2/9.693 deg. amperes. 

The vector representing this quantity is drawn to scale in the 
diagram, making an angle of 9.693 degrees with OEq. The result- 






120 ALTERNATING CURRENT PHENOMENA 

ant generator current for 100 amperes load current at unity 
power-factor is given then by the vector sum at 01, and is 

I = 128.3 /58.8 degrees amperes. 

By a method identical with that followed for the voltage diagram, 
the generator current for any value of load current at unity power- 
factor is obtained by taking the proportionate value of the vector 
AI. If the load power-factor changes, the vector A I is rotated 
through an angle equal to the power-factor angle, so that solutions 
for all values of load current between and 100 amperes are deter- 
mined by the intersections of the circular arcs corresponding to 
the particular numerical value of current and the line drawn from 
A making an angle equal to the power-factor angle with A I. The 
dotted lines are the vectors for a load current of 100 amperes, 
lagging 25 degrees behind the load voltage. 

77. Power-Factor from Vector Diagrams. — The power-factor 
angle at the generator is equal to the difference between the angles 
which E and I make with E . Thus, where E = 54,30 0/38.2 deg. 
and I = 128.3 /58.8 deg. the difference in the angles is 20.6 degrees. 
The angle of I is the greater, and thus the current is ahead of the 
voltage in time-phase position. The power-factor at the generator 
is cos 20°.6 = 0.936. The volt-amperes at the generator = ei, 
and the power in watts = ei cos 6. 

78. Operating Characteristics of Line with Various Loads. — 
In Table XV are collected a number of numerical values, taken 
from the vector diagrams by measurement, to illustrate the 
characteristics of the line under consideration. The values are 
all based upon a constant load voltage of 50,000 volts to neutral. 
Three power-factors at the load are considered, 25 deg. lagging, 
deg. lag, and 25 deg. leading. Since the numerical values are 
obtained graphically, the last significant figure may be slightly in 
error. The curves in Figs. 35, 36, 37, 38, 39, and 40 are plotted 
from the values tabulated in Table XV. 

As would be expected, the generator voltage increases most 
rapidly (to maintain E constant) in the case of the lagging load, 
and least rapidly for the leading load. In fact, the generator 
voltage nearly doubles in the former case, while in the latter case 
the increase is only 33 per cent for a range of 7 from to 100 
amperes. 



OPERATING CHARACTERISTICS OF LINE WITH LOADS 121 

K.V. 











































60 








No. 1, ~5 i Lagging Loin 
No. 2, Of {' I " 


. 
























58 








No. 


f, ts 


Leading 


IjOUc] 


























56 










































54 




























1 














52 










































50 










































48 










































46 
























\s 


















44 










































42 










































40 






















J*> 




















38 










































36 










































34 


















































32 






( 




































30 



10 



20 



80 



90 



100 



30 40 50 60 70 

Load Current, I , in Amperes. 

Fig. 35. — Generator Voltage Required to Maintain E Constant at 

50,000 Volts. 

150 





„ 




































^< 


.mp, 










































110 






Ncl. 1, 25" I 
Nd.2, |o° 


Bffg 


uk' I 


-.oail 
































No 


.3,2 


l»l 


eadi 


ngl 


oad 












8 


>X 












130 




















































































120 




























2 
























































110 




















































































100 




























_J_ 
























































00 




















































































80 























































































Pig. 36. 



20 



80 



;>o 



100 



80 40 50 60 70 

Load. Current, Jo » in-Ampcres. 

Generator Current for Different Load Currents with E 
Constant at 50,000 Volts. 



122 



ALTERNATING CURRENT PHENOMENA 











































80 








































Dq MM 










































70 




















































































60 


























































^N 




^1 






















60 


















:1 s 


































































40 




No 

No 


1,2. 

2. 


fit 




«x 


iaii. 
































No 


3.2 


,'L. 


•ailii 


1^' i- 


jad. 




























30 
















































































^ 




20 




















































































10 
























































































10 



■ 



SO 



M 



100 



30 40 50 60 70 

.Load Current, 1 , in Amperes. 

Fig. 37. — Angle by Which Generator Current Leads Generator Voltage. 
Load Voltage Constant at 50,000 Volts. 




10 



30 



BO 



BO 



30 40 50 60 70 

Load Current, I , in Amperes. 
Fig. 38. — Kilovolt-amperes at the Generator with Load Voltage Constant 

at 50,000 Volts. 



OPERATING CHARACTERISTICS OF LINE WITH LOADS 123 































































































No. 
No. 


1. 25 
L 


Lagging 

"1 


Loai 


































No. 


\ 25 


Lea 


ling 


Leai 


. 


























































3 


/\ 












































































































































































































































































































































































































I 


oat 


Cu 


rren 


M 


, in 


An 


per 


3S. 

















10 


20 

1 


T 


f 


50 


T 


f 


80 


T 


lOo 



K.W. 

7000 



6000 
5000 
4000 
3000 



2000 



1000 



Fig. 39. — Power at Generator for E Constant at 50,000 Volts. 














































80 


















h-* 
























i 




















2 






















70 




















T~" 






























































60 




















































































50 




















































































40 
















No. 

No. 


1,26" 

'-. 


Lagging Load 

" r- 


































No. 


i, 25 


Lea 


ling 


Load 


















L 




















































































20 




















































































10 








































- 
















































10 



30 40 50 60 70 

Load Current, I ft , In Amperes. 



itO 



100 



Fig. 40. — Efficiency of Transmission for E Constant at 50,000 Volts. 



124 



ALTERNATING CURRENT PHENOMENA 



> 
X 

w 



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E 

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§ 5 



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— i CM CO 'O CO 



35' 



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CN-H 


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288SS8 

in 35 cm cm r- co 



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co 

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«5 i-i CM CO t^- CO 

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oodddd 



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0.780 
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r-i in <- < oo oo oo 
oo co in co cm i-i 


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84.7 

79.4 

74.8 

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68.0 


9.693 
19.3 
28.2 
36.1 
43.2 
49.2 



CNiCOXO 



CONSTANT GENERATOR VOLTAGE; RIGOROUS SOLUTION 125 

In Fig. 36, showing the generator current, it is seen that at a 
load power-factor angle of 25 degrees lag, the generator current is 
practically constant throughout the 100 amperes range of load 
current. The total range of generator current is only from 94.2 
amp. to 102.9 amp. — a maximum variation of 9.2 per cent. At 
this condition of load, the result of the power transmission is to 
effect a constant current to a constant voltage transformation. 
For in-phase and leading load currents, the generator current in- 
creases quite uniformly, though by no means as rapidly as the load 
current. As shown by Fig. 37, within the range of 25 deg. lag to 
25 deg. lead, the generator power-factor depends only very slightly 
upon the load power-factor, but does depend upon the load current. 

The transmission efficiencies as shown in Fig. 40 reach a maxi- 
mum value in all three cases at a load current of from 70 to 80 
amperes. As would be expected, the efficiency is greatest for the 
lagging load — the PR losses in the line being greatly reduced on 
account of the diminished resultant value of line current, the 
lagging component of the load current neutralizing, in part, the 
leading capacity current of the line. Of course, if the leakage 
losses in the line were considerably greater (increased conductance 
g), the increased E 2 g loss, on account of the greatly increased 
voltage required at the lagging load power-factor, would cause the 
efficiency in this case to be lower, and in extreme cases, lower, even, 
than the efficiency at unity and leading power-factors. 

79. Constant Generator Voltage ; Rigorous Solution. — In 
the preceding section we have discussed a very convenient method 
for determining the performances of a transmission system when 
the - load voltage and load current are both known — the con- 
venient graphical processes being based upon the explicit equa- 
(19). If the problem be stated differently — say constant gen- 
erator voltage and known load currents — it would appear that 
no such convenient solution is possible. The knowns in such a 
case are load current and power-factor, and generator voltage. 
We are not, in fact, at liberty to specify both the load current 
and generator voltage throughout unlimited ranges, for we might 
select a value of load current greater than the generator voltage 
could supply over a short-circuit at the load, even. The second 
of equations (19), Chapter 2, when written 

7o = Z coshV7 + C/sinh VV (21) 



126 ALTERNATING CURRENT PHENOMENA 

to conform with the notation used in the alternating current 
problem, can be made to yield a solution, though not without 
involved expressions and tedious numerical operations. The 
procedure is as follows: the numerical values of E and J are 
known as scalars; e being the fixed generator voltage and i the 
value of the load current for which a solution is desired. The ratio 
of the two components of the effective load impedance Z is 
known, for it is this ratio which determines the load power-factor. 
The numerical value of '/... is unknown, and its determination con- 
stitutes the essential part of the problem, viz., the determination 
of the load voltage and the generator current. 
Let 

<fo = tan -1 — • 

Then the load current lags by the angle <£o behind the load voltage, 
and 

Zo = 2o cos <£o + jz<> sin <£o = Zo/^o, (22) 

where Zo is the unknown scalar value of Z . With Zo retained as an 
unknown and the angle <fo known, the product 

Z cosh VI = Zo (a + jb), 

in which a and b are known, and also (23) 

UsmhVl = c+jd 
is known, so that 

Zo cosh VI +U sinh VI = Zoa + c + j (z b + d). (24) 

E 
By (21), the scalar value of j- is equal to the scalar value of the 

■to 

expressions in (24), so that 



(ff-(**+c)*+W>+dp, 



or 



zo 2 (a 2 + V) + zo (2ac + 2bd) + c 2 + & -(^\ = 0, 
which gives 



- (ac + bd) ± 
20 ~ a 2 + 6 s 



y (ac + bd) 2 - (a 2 + 6 2 ) c 2 + d 2 - (£j 



(25) 



CONSTANT GENERATOR VOLTAGE 127 

The positive value of z from (25) is to be retained. Failure to 
secure such a positive value indicates that a value of t'o has been 
used which is greater than the generator voltage e could supply. 
Having the numerical value of z , the load voltage is immediately- 
obtained by 

E = Zo (cos <£o + j sin <fr>) 7 o, 

where 7 may be used as reference vector. Then E and 7 are 
obtained by equations (19) in their proper complex quantity 
notation by using 7 and E as found. A good check on the 
accuracy of the numerical work is afforded by the fact that the 
recomputed scalar value of generator voltage must agree with the 
value assumed at the outset. Of course, it is not necessary to 
recompute the value of E unless it is desired to check the work 
or to secure values of generator power, in which case it is neces- 
sary to do so in order to secure the power-factor angle at the 
generator. 

80. Constant Generator Voltage ; Solution by Use of Plotted 
Curves. — The procedure above outlined is not at all adapted 
to the computation of data for curve plotting, but it, or its equiva- 
lent, must be followed if results more accurate than could be read 
from curves are desired. If plotted curves are sufficiently accu- 
rate, then the solutions for constant generator voltage may be 
obtained with great ease by an indirect process, based upon the 
results obtained for constant load voltage in the preceding para- 
graphs. 

By equation (21) we see that 7 varies directly as E as long as 
the equivalent load impedance Z remains constant. E = ZoIq 
thus varies directly as E also. In Table XV, each solution as 
tabulated for a given value of 7 corresponds to some particular 
value of Zo, so that if we assume that the value of Z remains 
constant while the generator voltage changes from the tabulated 
value, obtained by the graphical solution described, to the constant 
value we wish to consider, the corresponding values of 7 and E 
for this latter case can be found by taking them in the same ratio 
to the original values as the ratio of the new E to the original E. 
The generator current will also change in the same ratio, while the 
values of power and volt-amperes will change by the square of 
this ratio. For example: in the solution for 7 = 80 amperes at a 
leading power-factor angle of 25 degrees and Eo = 50,000 volts, 



128 ALTERNATING CURRENT PHENOMENA 

E = 42,250 /43.2 dcg. and / = 136.9 /71.2 deg. If E be changed 
to 50,000 volts, then by proportionality 

7 = 4225? X 136,9 = 162 ^P 61 " 68 . 

50 000 
h = aoo-o x 80, ° =94 - 7 amperes. 

E = |^^ X 50,000 = 59,200 volts. 
Po =(^^) 2 X 3625 = 5080 K.W. 

p =(ti5) 2x5100 = 7140K - w - 

K.V.A. = (|f^) 2 X 5780 = 8090K.V.A. at gen. 

The efficiency, of course, does not change, since both load and 
generator powers change in the same ratio. It may be remarked 
here, as will be discussed later, that the efficiency of transmission 
of a given line depends only upon the equivalent load-end imped- 
ance Z , and not in any way upon the magnitude of voltage applied 
at the generator end of the line. 

Thus, in general, let 

K _ new generator voltage . . 

former generator voltage 
Then 

New currents or voltages = K (former currents or voltages), 
New powers or volt-amperes = K 2 (former powers or volt-) (27) 

amperes). 

In the transmission line under discussion let the generator volt- 
age be so adjusted that the load voltage is 50,000 volts at a load 
current of 80 amperes, lagging 25 deg. Its value from Table 
XV = E = 54,800 volts. Table XVI shows the method of ob- 
taining the data for performance curves under the condition of 
constant generator voltage from the data for constant load voltage 
in Table XV. 

The curves in Figs. 41 to 46 inclusive have been plotted with 
load current as abscissse, or independent variable, from the data 
in Table XVI. 



SOLUTION BY USE OF PLOTTED CURVES 



129 











































Lmp. 










































180 
























Ji_ 


















170 






















































160 










































150 
























2 


















140 










































130 










































120 
























^1 


















110 






No. 


l,2.-> 


"La 


ersir 


?L 


5aci 


























100 






No. 

No. 


2. 0« |" 
3,25°LeadlE 


gr Load 


























.90 










































80 










































70 










































60 
















Lot 


dC 


irre 


at I 


o,i 


nA 


mpc 


res 












50 





1 


3 


_J 


3 


3 





* 


3 


5 





61 | J 


3 


8 





9 


° 


1(0 



Fig. 41. — Generator Current for Generator Voltage Constant at 54,800 Volts, 









































Kilo- 








































< 


rolts 
80 




















































































75 










































































^8 










70 




















































































65 




















































































60 
































\2 
















No. 
No. 


1,25 

.'. 


;La K 




T.o;i( 


. 


























55 






No. 


3,25 


Lea 


ling 


I. OP. 


I. 


























































\I 










50 


























































Los 


dC 


ilT( 


nt, 


Co, 1 


i A 


ape 


res. 












45 





T 


I 


30 


o 


T 


" 


ik 


8. 


" 


100 



Fig. 42. — Load Voltage with Generator Voltage Constant at 54,800 Volts. 



130 



ALTERNATING CURRENT PHENOMENA 











































































































T"~ 




























































































































:.' 




















































































































































































































































No. 

No. 


i". 


uig 


ring 


Load 


































So. 


{U 


u 


ling 


1,oiW 


. 








































































Lo 


id c 


urr 


?nt 


I... 


n^ 


mp< 


2rcs 



















1 


1 


" 


T 


f 


60 


T 


i 





EI 


T 


wo 



K.V.A. 

10,000 



9,000 
8,000 
7,000 
6,000 
5,000 
4,000 
3,000 



Fig. 43. — Kilovolt-amperes at Generator for Constant Generator Voltage 

of 54,800 Volts. 




10 20 30 40 50 60 70 80 90 100 

Load Current, I , in Amperes. 

Fig. 44. — Power at Generator for Constant Generator Voltage of 54,800 Volts. 



SOLUTION BY USE OF PLOTTED CURVES 



131 











































i.w 

7000 


















































No 
No. 


1, 25 " LafjBiut' Load 

2. Q° | S | P 
























6000 








No. 


3, 2E 


»& 


adii 


gL( 


xul 


























































3^ 








50O» 


































S-- 


















































1000 


































1 






















































3000 




















































































3000 




















































































1000 






















































Lpad,< 


Sir 


•out 


I„ 


iD / 


Lm 


sere 


9 


















T 


20 

1 


T 


40 

1 


50 


60 


T 


80 


T 


100 



Fig. 45. — Power at Load End with Generator Voltage Constant at 
54,800 Volts. 











































Jl 




















1 „ 




~~~^ 


















7R 










































,70 






















i— 
















■- 














































60 




















































































50 




















































































40 




















































































30 














No. 1, 25 
No. 2, 


Lag 


[tag 


Loai 


. 
































No. 


!, M 


'Lea 


1 1 rifjr 


Loft 


. 


















20 




















































































10 























































































I 


1 


1 


1 


) 


1 


) 


4 


) 


a 


) 


a 


i 


7< 


I 


81 


) 


n 


i 


K 






Load Current, I , in Amperes. 
Fig. 46. — Transmission Efficiency with Generator Voltage Constant at 

54,800 Volts. 



132 



ALTERNATING CURRENT PHENOMENA 



1 

O 

> 

I 



-r 
9 

s 



■n 



X 

3 

< 



i 

3 

X _ 
JS o 

.fi > 

H 8 

£ II 

■Sbeq 
« . 

"3 



3 tX 



■ 



o 



.a ^ 

8 a 

O O 
_ f* 



a 



CJ 



3 °" 

a. a 

■9 > 



* 



« ;£ 



&a 



3 



o 



"9|5q 

II 



-^(NrJ(«0«0 



Soo c-i ci >o 
»-iC* co co co 



COQQOQO 
00 »ft lift C5 Q ^h 



t~05O5O5 00 CO 

"J N °2 *2 92 "- 1 



:s: 






005«MN<0 



00 C5 CO O Q ^h 



28SS 

LOIN CO i 



c. r ■/■ r r 



OOQOOO 

co 35 -h t- o 

<N CO-* ■* "3 



ift i-H t» -^ c5 -* 
00 00 t~ t>. t» t» 



HO— <i 



NNNNO 



00500C5 

(ONHtDHft 

■ft ■* ■«* co co oi 



t^-QOO^O 

OOOMhO-h 



!38 

ii»eo 



i-ioocooeo 
Ooot^o^o 



188° 



'OQ-JONO 
/: _. — ~. C". Sa 



,s 



1-1 Ift 05 1^00 



OOiftOCOOO 



b- >ft o oo co co 



'—' o 5 © SB i 

ift <M 55 iM (N I 



INFINITE SERIES TRANSMISSION LINE SOLUTIONS 133 



81. Infinite Series Form for Transmission Line Solutions. — 

For short lines operating at low frequencies and having only 
moderate power losses, the solutions as given in equations (3) and 
(19) may be expressed by infinite series, the successive terms in 
which decrease rapidly. Equation (19) is perhaps more easily 
expanded than equation (3), so the former will be used. The 
series for the hyperbolic cosine and sine are 



coshz=l+g-+g4-^+. ' 

/y»o /j*5 T** 

sinhz = z + [ 3+g + g-+-. 
Introducing these series in (19) with 

x - VI = VzYi = Vziv~t, 



(28) 



where 

and 

we have immediately 
Z t Y t , Z?Y t * 



= #o(] 
= /o(l 



2 
Z t Y t 



2A 
Z?Y? 



Z t = Zl, 
Y t = Yl, 



••W5(- 



%YA 



(Z t Y t )i 



6 



• -Wit^-^ 



(29) 



2 ' 24 

which, by combining factors in the second portion of each, become 
Z t Y t 



E 






2 ' 
+ hZt(l + 
Z t Y t 



Z?Y? Zt z Y t 3 



24 ' 720 ' 
Z t Y t , Z?Y? 



Zt z Y t 3 



G 



120 



2 



9 Y t ( 



z?y? zn? 

24 "*" 720 + ' 
. ZtYt Z?Y? 



G 



120 



5040 

) 

Z t 3 Y t * 
5040 



+ 



)] 



(30) 



Of course, Z t and Y t must be used in their complex form. The 
rapidity of convergence of these series is dependent upon the value 
of the argument Z t Y t) and thus upon the square of the product 



134 ALTERNATING CURRENT PHENOMENA 

of the line length and frequency, and also, to a lesser extent, upon 
the product of total line conductance and total line resistance. 

The form of any type of solution is worthy of attention. The 
generator voltage, or current, is always expressed in terms of the 
load voltage and current by linear equations of the form 

E = AE + BIo,\ 
I = Ah + CE , J 

in which the constants A, B, and C are functions of the physical 
properties of the system and the frequency only, and not of the 
voltages or currents. In effect, all the various methods for trans- 
mission line solutions are only schemes for determining the values 
of these three constants. Having them, the solutions for various 
loads are obtained with as much ease in the case of a long line as 
in the case of a short one. In equation (31) the functions 

cosh VI, U sinh VI, and jj smn ^ 

are the values of A, B, and C, respectively. In equation (30), the 
series are again merely expressions for these constants. It may 
be remarked here that the current and voltage at any point in a 
complicated network of circuits containing but one source of 
energy can always be expressed as such linear functions of the 
voltage and current at any other point, as long as the physical 
properties of the different parts of the system are pure constants 
and independent of the voltage or current. By far the most 
advantageous procedure in the calculation of the performances of 
any such system is to calculate first the values of these constants 
and then to apply the resulting numerical expressions to the 
particular numerical values of E Q and 7 for which a solution is 
desired. Further, expression in this form always leads to the 
convenient graphical process of calculation as just described in 
application to the long transmission line. 

82. Illustration of Convergence of Series. — To illustrate the 
manner of convergence of the above series expressions for A, B, 
and C, the following table shows their values, for the 400 mile line 
under consideration, when different numbers of terms are included 
in the parentheses. 



LINE ADMITTANCE AND IMPEDANCE LOCALIZED 135 



TABLE XVII 

Convergence of Series Expression for the Constants A, B, and 
C in Solution for 400 Mile Transmission Line 



Number 

of terms 

used 


A 


B 


C 


1 


1.00000/0°. 000 


326. 70/70°. 324 


0.0022024/88°. 439 


2 


0.67732/11.093 


290.53/73.124 


0.0019586/91.239 


3 


0.69035/ 9.652 


291.52/72.927 


0.0019653/91.042 


4 


0.69020/ 9.694 


291.51/72.931 


0.0019652/91.046 


Inf. 


0.69020/ 9.693 


291.51/72.932 


0.0019652/91.046 



An inspection of the tabulated values shows that, as far as 
engineering calculations based upon the more or less uncertain 
values of the line constants are concerned, the use of three terms 
in the series expression yields results, in this case, which are 
sufficiently close to the true values as given by the use of the 
hyperbolic functions (infinite number of terms). In the case of 
the B and C coefficients, two terms give very accurate results, but 
on account of the slower convergence of the hyperbolic cosine 
series, it is necessary to use at least three terms in the calculation 
of the coefficient A. 

In spite of the rapid convergence of these series expressions, the 
amount of labor involved in their use is much greater than for 
a rigorous solution based upon the hyperbolic functions directly, 
as in equation (19), provided tables of hyperbolic functions of 
real variables are available. Such being the case, there is no real 
need of setting up criteria for estimating, before the numerical 
computation is undertaken, the number of terms which must be 
used in the series to give a desired degree of approximation. The 
writer is well aware of the aversion of the general engineering 
public to the use of such hyperbolic function forms of expressions 
as equations (19), but, when such expressions are so easily dealt 
with numerically and at the same time furnish results concerning 
which there can be no doubt as to the accuracy, it appears that 
such prejudice is unwarranted. 

83. Line Admittance and Impedance Localized. — From 
purely physical considerations, it appears that the effect of the 
distributed admittance of a transmission fine can be approximated 
by placing a lumped admittance across the fine at the mid-point 



136 



ALTERNATING CURRENT PHENOMENA 



of its length, equal in value to the total amount of the distributed 
admittance. Doing this, the equivalent circuit representing the 
transmission line is as in Fig. 47. 
In this circuit, 

E' = voltage at mid-point = E + 5 ZJ , 
I' - current through admittance = E'Y t = Y t E + £ Z t YJ . 
I = I + I' = I (l + lZ t Y t ) + E Q Y h ) 



W^W^r^VV^^ 



Fig. 47. — Approximate Representation of Distributed Line Admittance. 

In equation (32), based on the approximation that the admittance 
of the line is all concentrated at the middle, the constants in the 
general form of equation (31) have the values, 



A = {\ + \Z t Y t ), 
B = Z t (l + \Z t Y t ), 
C=Y t . 



(33) 



Comparing these values with the series, equation (30), we see 
that the expression for A consists of the first two terms in the 
series for the hyperbolic cosine. The expression for B is also the 
first two terms in the series for U sinh VI, except that the coeffi- 
cient of Z t Y t in the parentheses above is one-fourth, instead of 
one-sixth, as in the true series. The expression for C in (33) is 
equivalent to the series expression in (30) when the first term only 
(unity) is retained in the parentheses. The amount of compu- 
tation involved in using equations (33) is the same as in using the 
first two terms only of the rigorous series expressions (30), and the 
results are not as accurate, because the values of the coefficients 
in the B and C terms are not the same as in the series. Therefore 
there is no possible excuse for using the approximate expressions 
based upon a concentration of the total admittance at the middle 



LINE ADMITTANCE AND IMPEDANCE LOCALIZED 137 

of the line, except, perhaps, in an academic way, to impress upon 
students the physical significance of leakage and capacity effects. 
Another approximation method is based upon a different con- 
centration of the line admittance. It assumes that the distributed 
line admittance may be replaced by locating two-thirds of the total 
admittance at the mid-point of the line, and one-sixth of the total 
at each end. This method is equivalent to assuming that the 
electrical quantities are distributed along the line in a way repre- 
sented by an arc of a parabola. (See Steinmetz' "Alternating 




W^W—r- W^fl^ 





Fig. 48. — Approximate Representation of Distributed Line Admittance. 



Current Phenomena," page 225.) Figure 48 shows the arrange- 
ment. Using the notation shown in the diagram, 

E 1 = E + IihZ t = Eo(l+f s Z t Y t ) + i ZJ . 
I" = | Y t E x = E Y t (f + ft Z t Y t ) + i Z t YJo. 
h = h + I" = E Y t (f + A Z t Y t ) + 7o (1 + | ZtYt). 
E^Ei + iZJ*. I = h + \Y t E. 

Zt 2 Y t 2 \ , r „ L . Z,Y t > 



E 



= £o(l+^-t 



2 

ZtYt 



36 



}+«* + ¥9 



r-z.(x + ^ + ^)w,(x + |z.r. + ^)-| 



(34) 



Naturally, equations (34) are closer approximations to the 
series in equations (30) than are those developed on the assumption 
that the line admittance may all be concentrated at the middle. 
In the above equation, the expression for the coefficient A differs 
from that in the series by only 



Z?Y? 
72 



higher power terms, 



138 ALTERNATING CURRENT PHENOMENA 

and is therefore quite accurate. The expression for the coefficient 
B does not contain the term in ZfYf, nor any above this order, 
but the coefficient of the Z t Y t term is the same as in the series. 
In the expression for the constant C the coefficients of both the 
Z t Y t and the Z t 2 Y t 2 terms are smaller than in the series. The labor 
involved in computing from equations (34) is the same as from the 
series expressions, when the same number of terms are included, 
so there is no reason for using the former, since they are not as 
accurate as the series when the latter are written to the same 
number of terms. 

To summarize, it is not possible to find any method of obtaining 
solutions for the type of alternating current transmission lines 
under consideration which surpasses, in ease of application, con- 
ciseness of expression, or in accuracy of results, the general and 
rigorous formulae expressed in terms of hyperbolic functions. 
Even the simplest approximation — single equivalent admittance 
at the middle of the line — does not materially lessen the work, 
and, as pointed out, is quite inaccurate — absolutely so when the 
line approaches a quarter-wave length. 

84. Collection of Formulae for Use in Computing. Tabular 
Arrangement. — For the convenience of computers, the formulas 
and tabular form for the computation leading to the determination 
of the constants A, B, and C, in the general form of solution (31) 
through the use of the rigorous hyperbolic function expressions 
(19), are collected here. 



COLLECTION OF FORMULAE FOR USE IN COMPUTING 139 

Collection of Formula 

r = line resistance per unit length, ohms. 

g = line leakage per unit length, mhos. 

L = line self-inductance per unit length, henrys. 

C = line electrostatic capacity per unit length, farads. 

/ = frequency in cycles per second. 

I = line length in units used above. 

x = 2 wfL. Z = r+jx = z(6,. 



1/ = tan- 1 -. y = -^— = -^- = Vg^+¥. 



fl^tan" 1 -. z = ^— = -^-= Vr 2 + 
r cos B z sin 6 S 

b = 2irfC. Y^g+jb^y/dj,. 

9 & 

y = — - — = 

cos 0„ sin By 

V = VZY = v/6v = a+jp. v = Vzy. 20 V = 0, + V . 

a = v cos 0„. jS = v sin 0„. 

c/ = y| = M/0 Jf . u = \J-. 20 tt = 0*-0„. 

A = cosh VI = cosh al cos /3Z + j sinh al sin 01 = a/6 a . 
S = sinh VI = sinh a£ cos 01 + j* cosh al sin j8Z = s/0,. 

(Express cosh VI and sinh VI in the form of scalars with 
their angles in the same way as was done for the im- 
pedance Z and admittance Y above). 

B = U sinh VI = 6/06. b = us. 6 b = U + 0,. 
C = ^ sinh V7 = c/0 c . c = -. C = 0. - 0«. 

The values of the three constants A, B, and C, to be used in the 
general form of equation (31), are obtained by the above formulae. 
Of course, the symbol C as used in these equations is not to be 
confused with the same symbol when used to indicate electrostatic 
capacity. 






140 



ALTERNATING CURRENT PHENOMENA 



TABLE XVIII 
Tabular Form for Computing from Formula on Preceding Page 



r 


6.560 


log cosh al 


0.16919 


i 


o.iooxio-^ 


log sinh al 


0.03588 


0.003788 


log COS lil 


+9.88128 


C 


0.00790 X 10-« 


log sin 01 


-9.81223-10 


f 


800 cycles 






I 


200 miles 


log cosh al cos 01 


+0.05047 


log2x 


0.79818 


log sinh al sin fil 


-9.84811-10 


log/ 


2.90309 


log tan d a 


-9.79764-10 


log L 


7.57841-10 


log cos 6a 


9.92790-10 


log 2 rf 
logC 


3.70127 






1.89763-10 


A = S log a 


0.12257 


log X 


1.27968 


cosh VI \ 6a 


327.890 deg. 


log r 
log tan d g 
log sin 6, 

log 2 

e, 


0.81690 
0.46278 
9.97564-10 
1.30404 
70.990 deg. 


log sinh al cos 01 
log cosh al sin 01 
log tan "., 
log sin 6, 


+9.91716-10 
-9.98142-10 
-0.06426 
-9.87924-10 


log b 


5.59890-10 


sinhWJ 10 ^ 


0.10218 


log fir 


3.00000-10 


310.777 deg. 


log tan By 


2.59890 






log sin By 


0.00000 


B = )log b 


2.95475 


log y 


5.59890-10 


U sinh VI \ 6b 


301.344 deg. 


0y 


89.856 deg. 


C =jlogc 


7.24961-10 


log V 2 
2 6 v 


6.90294-10 
160.846 deg. 


^sinh V1 1 C 


320.210 deg. 


e v 


80.423 deg. 






log cos 6 V 


9.22108-10 


Numerical Res ults. Eauation31 


log r 


8.45147-10 






log sin 6 V 


9.99390-10 


A 1.3261 /327.890deg. 


log a 


7.67255-10 


B 901.1 /301.344deg. 


log/3 


8.44537-10 


C 0.001777 j 


/320.210deg. 


log 360/2 t 


1.75812 






log 0° 


0.20349 






a 


0.0047049 


A minus sign before the logarithm 


0° 


1.59768 deg. 


means that the quantity represented 


at 


0.94098 


by the logarithm is negative. 


/3°Z 


319.536 deg. 






log 2/y 


5.70514 






2 u 


-18.866 deg. 






"i log l 


2.85257 






-9.433 deg. 







The numerical values entered in the table correspond to a line 
whose constants are those given in the first six entries of the table, 
and are approximately those for No. 8 B. & S. gauge wire spaced 
18 inches. See " Standard Handbook." Constants for loop of 
two wires. 



CHAPTER VII 

THE PROPAGATION CONSTANT, V = VZY = a + j(3 

85. General. — In the preceding sections it has been shown 
how to obtain the numerical values of the quantity V = a -f j(i 
which enters into the exact solution for a line carrying an alter- 
nating current. In Chapter VI it is seen that a, the real portion 
of V, is the term which accounts for the decrease in the magnitude 
of the component waves in the direction of their propagation. 
For this reason the name "Attenuation Constant" has been given 
to the quantity a. The quantity j3, as the imaginary portion of 
V, determines the amount of rotation of the vectors representing 
the component waves for each unit of line length. /3 is numeri- 
cally the rotation of each vector, in radians, per unit length. 
Since a complete rotation of one of these component vectors 
takes place in a distance along the line equal to one wave length, 
the constant /?, which determines this distance, is termed the 
"Wave-length Constant." The combination of the two quantities 
into the single complex, V = a + j/3, is known as the "Propaga- 
tion Constant," since this quantity completely determines the 
manner of propagation along the line of the separate component 
waves, and thus their resultant. Both components of V are de- 
pendent upon all of the line properties — resistance, self-induc- 
tance, leakage, electrostatic capacity, and frequency. 

86. Fundamental Equations for a and p. — The attenuation 
constant, a, depends, in the main, upon the line resistance and 
leakage, but its value is also affected by the other three properties. 

By equation (5), Chapter VI, 

a = y/\ (zy -xb + rg), (1) 

which, when expanded by 

x = wL, b = ojC, z = Vr 2 + x 2 , y = Vg 2 + 6 2 , 

becomes 

a = \/\ { V(r 2 + « 2 L 2 ) (g 2 + J&) + rg - o> 2 LC \ . (2) 

141 



142 THE PROPAGATION CONSTANT 

The quantity /3 depends principally upon the frequency, self- 
inductance, and capacity, though it is not independent of resist- 
ance and leakage. By equation (5), Chapter VI, 

= y/\ (zy + xb- rg), (3) 

which by the above expansion becomes 

$ = Vh WV^tfL?) (c? 2 + c^C 2 ) - rg + <**LC}. (4) 

The manner of variation of these two quantities, a and /3, with 
variation of the several separate quantities entering into their 
determination, has been extensively studied by Pupin, Fleming, 
Heaviside, and others, on account of their importance in deter- 
mining the operating characteristics of long electric cables. 

If / = 0, co = 0, and the expressions reduce to 

a] -V^. fi\ = 0, (5) 

Thus, V = y/rg + j = v as used in the discussion of direct 
J M =o 

current transmission over leaky lines, Chapter II. If r = and 
g — 0, we have immediately 

oL = 0; j8~L = Vxb = 2 irfVTC. (6) 

Jg=0 J ff =0 

87. Wave Length and Velocity of Propagation. — In any case 

the distance along the line required for a complete rotation of the 

component vectors is 

2x 
X = — = wave length, (7) 

P 

and the velocity of the separate waves is 

2irf 
S = /X = —^- = velocity of propagation. (8) 

P 

Thus, with no line losses (a = 0), 

xl = 2 * = 1 —, 
J«=o 27I-/VLC fVLC' 



and 



a.-s] - 

Jcr=0 



27T./VLC Vlc 



(9) 



SPECIAL CONDITIONS 143 

The velocity of propagation when there are no line losses is 1/VlC. 
This is the limiting value of the velocity, and is equal to the velocity 
of light = 3 X 10 10 cm. per second, for conductors in air, when the 
internal self-inductance of the metallic conductors is neglected. 
Except under one special condition, the introduction of energy 
losses due to resistance of the conductors or leakage between the 
conductors, will cause the value of S to be smaller than S x , 
although, for all values of r and g the velocity S at continuously 
increasing frequencies approaches S^ as a limit. This approach 
to S^ at infinite frequencies arises from the preponderance of the 
zy and xb terms in equation (3), so that 

*L=vfe= s - (l0) 

Also, at high frequencies, the internal self-inductance of the con- 
ductors diminishes, and finally becomes zero, so that at the limit, 
the velocity, £~L =a0 , becomes equal to that of light because of L 
becoming equal to the external self-inductance only. Of course, 
at these very high frequencies, the radiation of energy into space 
would introduce losses and thereby increase the effective values 
of r and g. Since equation (10) was set up under the assumed 
condition that r and g remain constant, it is, of course, not true in 
general when r and g are both functions of the frequency. If the 
product rg increases less than in proportion to the square of the 
frequency and L and C remain constant, then equation (10) 
remains true. The subject of radiation losses cannot be taken 
up here. 

88. Values of a, P, and S, when gL = rC. If equations (2) 
and (4) be expanded, 

«-ft lW + wW 2 + o>yL 2 + o> 4 L 2 C 2 )* + rg - <£LC }]*, 

= ft { W + <» 2 r 2 C 2 + «VL 2 + a> 4 L 2 C 2 )* - rg + co 2 LC |]*, 



(11) 



which, by adding and subtracting 2 ufrgLC to the term in paren- 
theses (see Fleming, "Propagation of Electric Currents," page 69), 
reduce to the following form: 



a = \/\ { V(gr + tfLCY + « 2 (gL - rC)* + (gr - rfLC) } , 
j 8 = V / i{V5r + co 2 LC) 2 + co 2 ^L-rC) l -(gr-co 2 LC) [.. 



(12) 



144 THE PROPAGATION CONSTANT 

The form in (12) permits of a more ready determination of the 
manner of variation of the values of a and /3 than those previously 
given, but as far as computing purposes are concerned it is open 
to the same objections as equation (5). If we assume that the 
fundamental constants r, L, g, and C do not change with the 
frequency (an assumption sometimes far from the truth), the 
manner of variation of a and /3 with the frequency can easily be 
determined by inspection of equations (12). 

If the relation gL — rC — be fulfilled, the equations for a and 
/3 reduce to 

a\ = V7g, /sl = oVZC, (13) 

JjL-rC JffL-rC 

indicating that a is independent of the frequency and /3 is pro- 
portional to the frequency, a in this case is equal to the direct- 
current attenuation constant, and the velocity of propagation 

JgL=rC VLC 

is constant, irrespective of the frequency. 

If the relation gL — rC = be not fulfilled, the second term in 
parentheses under the double radical in the expression for a has a 
value, either positive or negative, and since its square is always 
positive, the value of a will continuously increase with increasing 
frequency — from its direct-current value, Vrgf, towards some 
limiting value at infinite frequency. jS 2 increases continuously 
with increasing frequency, from at zero frequency towards 
(o^LC + constant) at infinite frequency. 

89. Infinite-frequency Values of a, P, and S. — The best way 
to establish the infinite-frequency values of a and /3 is to develop 

their equivalent expressions into power series in -, and then to note 

the finite terms when co = oo . Beginning with the fundamental 
equations (1) and (3), and letting 

- = w, r = mx, l = v > g = vb > ( 15 ) 

we have 

z = Vr 2 -!-* 2 = x (1 + u 2 )*, y = b (1+ v 2 )*, (16) 



, LOW-FREQUENCY VALUES 145 

which, when expanded by the binomial theorem and multiplied, 
give 

zy = xb (1 + \u 2 + h v 2 - } u 4 + I uh 2 - £ v* 

+ Aw 6 -tVwV- t VwV + 1 V« 6 + • • • )• (17) 
The fourth order terms in (17) may be combined into 

-\{u 2 -v 2 ) 2 
and the remaining higher order terms in the series under the symbol 

<A (u, v), 
so that 

zy = xb [1 + h 2 + i«" - I (w 2 - » 2 ) 2 + iA (w, *)]. (18) 

If this value of zy be substituted in" equations (1) and (3), for a 
and /3, 

a = V| rg + \ xb [u 2 + v 2 - \{u 2 - v 2 ) 2 + 2^ (w, v)], 



, (19) 
/3 = V- i rgr + \ xb [4 + u 2 + v 2 - \ (w 2 - v 2 ) 2 + 2^ (u, v)} 

The terms 

x6[-i(w 2 -y 2 ) 2 + 2^(w,y)] 

vanish at infinite frequencies because they are all of at least the 
second order in -. Therefore, on substituting for the various 

0} 

remaining quantities their expressions in terms of the fundamental 
line constants, 






(20) 



The curve for a, plotted in terms of the frequency, is asymptotic 
to the horizontal line at a height given by a] /=Q0 in (20). The 
curve representing /3 is asymptotic to the straight line passing 
through the origin, whose equation is /3 = 2 irj \ /r LC. As before 

mentioned, the velocity then approaches ■ r—^ as a limit. 

90. Low-frequency Values of a, P, and S. — To investigate the 
forms of the curves at very low frequencies, the expressions for a 

and may be expanded in terms of o> instead of — . Omitting the 

0} 



146 THE PROPAGATION CONSTANT 

details, which are much the same as in the previous case, except 
that we let 

x = Mir, z = r (1 + Ui 2 )*, b = Vig, y = g (1 + t>i 2 )*, 

the following series are obtained: 

a = V7g VI + \ ( Ml 2 + v, 2 ) - tV (u x 2 - t>i 2 ) 2 + * * (ui, t>0 

a =V r ^l+-_ + _- j, 

/8= Vi a* + i rg [u? + »i 2 - \ (ui 2 - »i 2 ) 2 + 2 * ( M| , „,)], 



(21) 



p = „y-LC + 5^— + ^ - B «^ -yj +. ... 

Thus the Umiting velocity of propagation at zero frequency is 

So = s] -§.- * (22) 

It is easily seen that the quantity in parentheses in (22) can never 
be less than 2 LC, so that the value of /3 is always greater than 
w VLC, except in the special case already mentioned, when gL = 
rC, for which the parenthesis becomes equal to 2 LC. In general, 
then, the velocity, S, is always less than the infinite frequency 
velocity. 

The two curves representing a and S as functions of the fre- 
quency start horizontally from their zero-frequency values, rise 
with increasing rapidity along curves approximately parabolic, 
reach their maximum slope at inflexion points where their second 
derivatives have a value of zero, and then, with continuously 
decreasing slopes, run out asymptotic to their infinite-frequency 
values. For fines with low resistance and leakage, the approxi- 
mately parabolic portions of the curves for low frequencies are very 
small in comparison with the large sweep of the curves in passing 
out to their final asymptotic values. With large losses, the curves 
flatten and approach their final limits more slowly. 

91. Illustration of the Variation of a, (J, and S with Frequency 
for Power Circuit. — The following table of values has been 
computed, to show the manner of variation of these quantities for 
the power transmission line previously considered, and whose con- 
stants per unit length are again given at the beginning of the table. 



VARIATION OF a, ft AND S 



147 



TABLE XIX 

Different-frequency Values of the Attenuation Constant, a, 

Wave-length Constant, in Degrees, 0°, and the Velocity 

of Propagation, S, in Miles per Second, for 

L = 0.002040, r = 0.2750, g = 0.15 X 10r*, C = 0.01460 X 10- 6 



/ 


a 


f 


s 





0.00020310 





94,007 


2 


0.00022583 


0.006888 


104,530 


4 


0.00025820 


0.012050 


119,510 


6 


0.00028454 


0.016401 


131,700 


8 


0.00030483 


0.020412 


141,090 


10 


0.00032070 


0.024253 


148,440 


12 


0.00033317 


0.028014 


154,210 


16 


0.00035098 


0.035456 


162,450 


25 


0.00037205 


0.052264 


172,200 


60 


0.00039077 


0.11942 


180,870 


180 


0.00039528 


0.35418 


182,960 


300 


0.00039566 


0.58973 


183,140 


420 


0.00039577 


0.82540 


183,180 


540 


0.00039581 


1.06111 


183,200 


660 


0.00039583 


1.29684 


183,210 


780 


0.00039585 


1.53259 


183,220 


900 


0.00039586 


1.76833 


183,220 


2000 


0.00039587 


3.9294 


183,230 


3000 


0.00039587 


5.8941 


183,230 


4000 


0.00039588 


7.8588 


183,230 


6000 


0.00039588 


11.7882 


183,230 


8000 


0.00039588 


15.7176 


183,230 


Inf. 


0.00039588 


Inf. 


183,230 



148 



THE PROPAGATION CONSTANT 











































V-f - 










































j«_ 
























































































































9 










































1 
























« 


















k 












































» 
























VI 


v\ 














o 








































^ 


co 








































a 

■ 

3_ 










































c 

E 


^! 








































~ 










































ei 




















































































© 





















































































2» 


















































































a- 








































| 

c 








































-I 


g 








































8. 








































I 

"3 


CO 








































5 










































x 


CO 








































■ 
- 












































-* 




















































































CM 




















































































o 



-I 3 



RELATIONS BETWEEN a, 0, AND S 



149 



The lower-frequency values entered in Table XIX are plotted 
as curves in Figs. 49, 50, and 51, in order to show the curve shapes. 
From an inspection of the table, it is seen that at 60 cycles, a and 
S have nearly reached their limiting values, and /3 is approximately 



s 


































x 10 
17 


































































16 


































































15 


































































14 


































































13 




































































































12 






































































































10 




































































































9 












































Fn 


que 


ncy 


/, < 


Jjrcl 


esp 


erS 


BOO 


id. 








8 







i 




4 




6 




8 


T 


1 


i 


1 


i 


\ 



Fig. 51. — The Velocity of Propagation as a Function, of the Frequency. 
(Circuit Constants Representing a Power Line.) 



proportional to the frequency. The infinite-frequency values of 
a and S are nearly twice their zero-frequency values. 

92. Relations between a, |3, and S. — The similarity between 
the curve representing a as a function of the frequency and that 
representing S is so striking that an explanation is needed. In 
equation (4a), Chapter VI, we have 



This may be written 



2 a/3 = rb + gx. 
2aP = w(rC + gL), 



(23) 



150 



THE PROPAGATION CONSTANT 



from which, in the expression for velocity, we have 

e _ u _ < ^ a 
^~/3~ rC + gL' 

. rC + gL 
P m — o w - 



(24) 



In the first of equations (24), rC + gh is constant, so that the 
curves for a and S as functions of the frequency differ only by a 
constant factor. Thus, knowing any one of the three quantities, 
a, j8, or S, the other two may be quickly determined by means of 
equation (24) without recourse to the more complicated general 
expressions for the separate quantities. 

93. Numerical Illustrations for Telephone Circuits. — To 
further illustrate the variation of a, /3, and S, the values entered in 
the following table were computed. The circuit constants used 
are approximately those for two No. 12, N. B. S. gauge copper 
wires spaced 12 inches — a standard type of aerial telephone 
circuit. The values of the constants are for a loop-mile, that is, 
wire to wire. 



TABLE XX 

The Propagation Constant and Velocity of Propagation in an 
Aerial Telephone Circuit of the Constants: 

r = 9.94, L = 0.00366, g = 1.5 XlO" 5 , C - . 00822 X I0r* 



f 


a 


0° 


s 





0.003861 


0.00000 


88,560 


5 


0.003874 


0.02026 


88,850 


10 


0.003909 


0.04015 


89,660 


20 


0.004035 


0.07780 


92,540 


30 


0.004203 


0.1120 


96,400 


50 


0.004573 


0.1716 


104,890 


100 


0.005381 


0.2917 


123,410 


150 


0.005967 


0.3945 


136,870 


200 


0.006392 


0.4911 


146,620 


300 


0.006942 


0.6783 


159,220 


400 


0.007260 


0.8648 


166,520 


600 


0.007582 


1.2423 


173,870 


800 


0.007725 


1.6255 


177,180 


Inf. 


0.007949 


Inf. 


182,310 



The above quantities are shown graphically in Figs. 52, 53, 54, 
and 55. 



NUMERICAL ILLUSTRATIONS FOR TELEPHONE CIRCUITS 151 











































s 










































s5 










































<T? 










































■>! 








































o 
B 
8 


g 








































u 

o 


1—1 








































►J. 
B 




















n_ 






















ft 








































*i 


1- 








































>> 
a 

a 








































a 








































£ 










































5- 


















































































o 











































d 




+3 




| 




Q 




o 




i 

3 




1 


a 


i> 


,c 


£ 


a 

O 




O 


0) 

,3 


H 


H 














































g 










































" 










































"? 








































-a 

s 

5 


H 








































o 
a> 


- 










































" 














«J 


























1 


s 








































>> 












































- 








































§ 










































0" 

f 










































£ 












































s 




















































































s 















































<j s 





















• 
















s 


































































160,000 
































































160,000 


































































140,000 




















Zy 














































190,000 


































































120,000 




















% 














































110,000 


























t 








































100,000 


































































90,000 


































































80,000 










Fr< 


■qu( 


ni'j 


./. 


Cyc 


>s i 


eri 


|60( 


ml. 













i 





T 


L 


D 


1&) 


200 

1 


1 


to 


280 

1 



Fig. 54. — The Velocity of Propagation along a Telephone Line. 



a 


p 










































0.007 


1.1 




































'/' 






1.3 












a 






% 
















v 


• 






0.006 


1.2 
































V 


' 










1.1 




























J> 


'• 

/ 












0.005 


1.0 


























»^ x- 


• 

X 
















0.0 
























-X- 


/ 
















0.004 


n> 
























/ 




















0.7 










|^ 










/ 






















0.003 


o.n 
















/ 


/ 


























0.5 














,/ 


/ 


























0.002 


o.t 












/ 
































0.:i 










/ 
































0.001 


0.2 






/ 


/ 




































0.1 




/ V 


/ 












































y 








F 


reqi 


lent 


y.f 


C>x 


les 


J'CT 


Soc< 


jnd 















Sx 

io-» 



200 



160 



120 



■HO 



40 



160 320 480 640 «00 " 

Fig. 55. — The Attenuation Constant, Wave-length Constant, and Velocity 
152 of Propagation for a Telephone Line. 



CHAPTER VIII 

THE EQUIVALENT IMPEDANCE OF CIRCUITS AND 

THE DETERMINATION OF THE FUNDAMENTAL 

CONSTANTS BY MEASUREMENTS 

94. General Expression for Generator-end Impedance. — 

The complex ratio between the generator voltage and the genera- 
tor current, when these two quantities are expressed in complex 
form from the vector diagram representing them, is the equivalent 
impedance of the transmission line, including whatever load im- 
pedance may be placed at that end of the system. The expres- 
sion for this impedance is set up in the same manner as for the 
analogous direct-current problem, viz., by taking the ratio between 
the general equations in (19), Chapter VII, for E and I in terms of 
E Q and To- The load-end impedance is the ratio between the load 
voltage and the load current. 

Z = -j^ = load-end impedance. (1) 

By substituting, E = Zolo in equations (19), Chapter VII, and 
dividing, the common factor 7 canceling from both the numerator 
and the denominator, we have 

Z cosh VI +U sinh VI 



Z a = 



cosh VI + jj sinh VI 



E 
= y = equivalent generator-end impedance. (2) 

In treating the subject of line impedances, it has been the 
custom of some writers to introduce separate names for the 
various ratios of e.m.f. to current, and these names are not, in 
general, such as to be self-explanatory. The introduction of such 
terms as "Initial sending-end impedance," "Final sending-end 
impedance," and lastly — the least comprehensible of all — 
"Final receiving-end impedance," which is not, as would be 

153 



154 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

supposed, the ratio between load voltage and load current, serve 
to confuse the mind of the reader. In this discussion we will 
confine ourselves, in general, to two symbols for equivalent 
impedance: Z = ratio between load voltage and load current, 
and Z g = ratio between generator voltage and generator current. 
The subscripts used are such as to indicate directly the meaning 
of the symbol. Both of the impedances, Z and Z g , are complex, of 
the form 

Zo = r + jxo, 

Zq — f g -\- JXg 



(3) 



95. Generator-end Impedance under Special Conditions. — 

Two equations arise from (2) as special cases, which are perhaps 
more important than the general form. They are the expressions 
for Z g when the load end of the line is short-circuited, and when 
the load end of the line is open-circuited. In the first case, 
Z = 0, and in the second case, Z = oo . Introducing these values 
of Zo and evaluating the resulting equations, 

Zg] = Z g ' = Ut&nh VI, 
.ta-o 

= short-circuited impedance. (4) 

Zg] = Zg" = £/cothW. 

= open-circuited impedance. (5) 

These two equations are analogous to equations (39) and (40), in 
Chapter II, to which they would reduce for / = 0. 

If, for the purpose of analysis, we assume that the line losses 
are zero (r = and g = 0), the equations (4) and (5) assume 
trigonometric forms because of a being zero. Thus, since 

tanhj/3 = jtan/3, and cothj/S = — jcot/3, 



(6) 



Z '\ =jU tan /3Z, 

Ja=0 

Z g f> ] = -jU cot pi 

Ja-0 

Again, when a = 0, the quantity U is no longer complex, but 

"L-vl-vl-v/1- <?> 



GENERATOR-END IMPEDANCE 155 

and by equation (6), Chapter VII, 

1 = co VIC, 

Ja=0 

so, finally, the equivalent generator-end impedances of a line with 
no losses, the load end being short-circuited and open-circuited, 
are respectively, 



zA =jy^ tan co VLCZ, 
Z/'l = -3\k cot uVZCl. 



(8) 



In such a system it is apparent that, for given values of L and 
C per unit length, doubling the line length at constant frequency 
will produce the same variation in Z g ' and Z g " as doubling the 
frequency with the original line length unaltered. This comes 
about from the fact that co and I enter the equations (8) as a product, 
and in no other way. The product VLCl may also be written 
VLtCt, where L t and C t represent respectively the values of the 
total line self-inductance and the total line capacity. In (8) the 
resulting impedances are either pure inductive or pure condensive 
reactance — no real component appearing in the equations. This 
must be obvious from physical considerations. If a real component 
appeared in the expressions for impedance, an energy loss would 
thereby be represented, which is contrary to the assumption that 
there is neither resistance nor leakage present. 

It is quite obvious that at low frequencies the short-circuited 
impedance of a line with zero losses will be inductive reactance 
and approximately equal to coL*, numerically. The first of equa- 
tions (8) reduces to this for low frequencies by neglecting all terms 
in the series expansion of higher order than the first in co. It is 
also obvious that for an open-circuited line at low frequencies the 
impedance will be condensive reactance and approximately equal 

in magnitude to -7=- . This is the first term in the series expansion 
coC* 

of equation (8), for Z". With increasing frequencies, neither of 

the two quantities, self-inductance and electrostatic capacity, can 

be considered alone in such a simple way, but their combined effect 

must be summed up in equations of the form of those in (8). 



156 



THE EQUIVALENT IMPEDANCE OF CIRCUITS 



96. Analogies between Circuits with Distributed and with 
Lumped Properties. — It is instructive to compare the results 
given by (8) for the impedance of circuits having distributed 
constants with the forms of expression for circuits with lumped 
constants, which, in an approximate manner, represent physically 
the short-circuited and open-circuited transmission line. In the 
case of a short-circuited line we may consider that the current 
flowing from the generator has, in effect, two alternative paths — 
one around the entire loop of the line, and thus through the entire 
amount of self-inductance, and the other through the line electro- 
static capacity. The phenomena in this case are approximated 
by those which take place in a parallel circuit, one path of which 
contains a condenser and the other an inductance coil. In the 



£ 



Short-oiixuited Liue 



L t C { 



£ 



Open-circuited Line 



Analogs. 



y W v 



Fig. 56. — Simple Circuits Analogous to Short-circuited and Open-circuited 
Transmission Lines. 



case of an open-circuited line, the analogy is different. Here the 
generator current may be supposed to flow through two parts of 
a circuit in series — a portion of the line constituting a self -in- 
ductive impedance and the electrostatic capacity of the two wires 
completing the circuit by a condensive impedance. If the capacity 
and self-inductance in the localized system be made equal, re- 
spectively, to the total capacity and self-inductance of the trans- 
mission line, as shown in Fig. 56, the curves giving the reactances 
of the different circuits appear as in Figs. 57 and 58, which are 
plotted with frequency as the independent variable for the 400 
mile line whose values of L and C are given in Table XII. Thus 
L t = 0.816 henry and C t = 5.84 X 10"* farads. 



ANALOGIES BETWEEN CIRCUITS 



157 



X 

.400 








1 j 
























/ 










L200 










at 
























/ 










L000 












































800 






l» 


1 




































600 






1 

1 






































400 






1 

1 
























a/ 














m 




/ 


/a 






































— 
■ 200 




/> 












Fi 


equenc 


vJ 

























4 





f 





|L 


.'0 


It 


•0 


a 


fei 


^2 


o_ 


_JB 


*? 


—a 


B— 


1-360— 


--400 


■400 












\< 


s> 


■"* . 




























• Ron 










/ 


A 

' J 




























/ 




■800 










/ 






























/ 




•000 










i 
i 




























/ 






•200 










(» 






a 






















/ 






L400 










/ 
i 


































-L600 






































1 







Fig. 57. — The Equivalent Impedance of a Short-circuited Line and 
of Its Analogous Circuit. 



1 x 
1400 






























A 


X 










L200 






















<•/ 






y^ 


-• 












L000 
























S 


''' 
















800 




















.</ 


K 




















600 


















,s' 
























400 


















'7 
























200 












/' 




























c/ 











t 


f 






F 


•('(JV 


cue 


r4 




















-200 





1 





A 

t 


)^> 


**\ 


D 


1 


iO 


a 





: 


!40 


% 


<0 


8 


J^- 


' 8 


D 


400 


■400 






/ 


/v 






















Qj 












-000 




i 


1 / 




































800 




It 

1 






































1000 




II 






































1200 


























c/ 
















1400 










































r 

1000 


\ 









































Fig. 58. — The Equivalent Impedance of an Open-circuited Line and 
of Its Analogous Circuit. 



158 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

The formulae for the simple analogous circuits are: 

1 uLt 



(Short-circuited line) Z = 



. _ .1 ' 1 - u*L t Ct 
J " C <- J ^L t 



(Open-circuited line) Z = j I wL« -^ J • 



(9) 



Numerically, the equations for the actual lines and the analo- 
gous circuits are: 

1. Short-circuited Line 

(a) Line equation, zA = j 373.8 tan 0.013716/. 

Ja = 

5 127/ 

(b) Analogue, Z = j ± , a000J8813/2 - 

2. Open-circuited Line 

(c) Line equation, Z a "\ = -j 373.8 cot 0.013716/. 

Ja-0 

(d) Analogue, Z = j (s.127/ - aQ000 3 5858/ ) 

The lettering of the curves in Figs. 57 and 58 corresponds to that 
in the above numerical equations. 

In both comparisons, the two curves in question coincide at zero 
frequency, but for frequencies above 20 cycles per second they 
separate rapidly. For the line, the first resonance frequency, as 
indicated by the infinite value of impedance for' the short-circuited 
line and by the zero impedance for the open-circuited condition, 
is 114.5 cycles per second. The 400 mile line is exactly one- 
quarter wave length at this frequency. This value of / may be 
determined by placing equation (a) equal to infinity. Then 
tan 0.013716/= inf., or 

0.013716/= (2»-l)|, /«fe^f, 

/= (2n- 1)114.5. 

The analogous lumped circuits reach their resonant condition at a 
much lower frequency than the transmission system — in this case 
at a frequency of 72.8 cycles per second. The ratio between these 



ANALOGIES BETWEEN CIRCUITS 159 

two frequencies, 114.5 and 72.8, is =, and this ratio holds true for 

any length of line, as may be shown in the following way. The 
first resonance frequency for the line occurs at a quarter wave 

length, or when fil = ~ . The resonance frequency in the lumped 

circuit occurs when the inductive reactance is equal to the con- 
densive reactance, numerically. Thus, for the line at the first 
resonance condition 

2 4 VL t Ct 

and for the lumped circuit 

2rfL t = ^, f = 



2*fC t ' J 2-K^L£ t 

The ratio of line frequency to lumped-circuit frequency, at reso- 

. - 2i 7r 
nance, is thus -r- = -• 
4 2 

In both Fig. 57 and Fig. 58, it is seen that the lumped-circuit 
impedance is equal to the line impedance at frequencies other than 
zero — that is, for the frequencies at which the respective curves 
intersect. In this case, at approximately 200 cycles, the imped- 
ances are again equal for both the short-circuited and open- 
circuited conditions. 

The values of the frequencies for equal values of impedance may 
be determined by equating the respective expressions. For the 
short-circuited line and its analogous circuit, from equations (8) 
and (9), for equal impedance, 

\/^tan c VLA = , . «fr , (10) 

and for the open-circuited line, from the same equations, 

5-.von--i.-fc 01) 

w coC< 

Equation (11) reduces to (10) by taking the reciprocal of both 

members and then multiplying throughout by y^, which shows 

w 

that the impedances in the respective cases become equal at the 

same frequencies. These transcendental equations, (10) and (11), 



160 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

are incapable of direct solution for u, which must, therefore, be 
determined by methods of approximation or by taking the inter- 
section of curves. 

The equations and curves for the equivalent impedances of a line 
with zero losses and with distributed self-inductance and capacity 
are of importance in determining the free frequency of oscillation 
of such circuits, and, in this connection, will be referred to again. 

97. Limiting Conditions at Infinite Frequencies. — In actual 
lines, resistance and some leakage are always present, so that the 
ideal conditions just considered are never realized exactly. In the 
actual case, the resulting impedance does not consist of reactance 
only, which reaches alternately infinite and zero values according 
to tangent or cotangent trigonometric functions. The energy 
losses in the line require a real component in the impedance 
expressions for short-circuited and open-circuited lines, and prevent 
the reactance curves from reaching infinite values, though they 
may pass through zero values. The real component of the 
resulting impedance can, of course, never be negative, for this 
would imply an average flow of power from the line into the genera- 
tor, and such a condition is impossible, if the line be either short- 
circuited or open-circuited as assumed. However they may vary, 
then, the angles of the two quantities 

Z g ' = U tanh VI and Z " = U coth VI 

can never atyain the limiting values +90° and —90° no matter 
how long a line or how high a frequency be considered. 

Since the attenuation constant, a, approaches a finite limit at 
very high frequencies, and the wave length constant, /3, approaches 
proportionality with the frequency, it may be shown that, at very 
high frequencies with a line of fixed length, the curves representing 
the equivalent resistance and equivalent reactance, for both Z ' 
and Z g ", approach strict periodicity, with frequency as the inde- 
pendent variable. At very high frequencies, the quantity U — 

y y. approaches the constant y ^ in value. The imaginary portion 

of U disappears on account of the preponderance of the reactance 
and susceptance over the resistance and conductance, respectively. 
By equations (20), Chapter VII, at infinite frequency, 

a = a x = c onstant, 
= o>VLC, 



INFINITE-LINE IMPEDANCE 161 

so that 
and finally, 

zj\ - y ^tanh (la* +j2rfVLCl), 

Zff "] ." vi coth **- + j 2irf ^^ 



(12) 



At these frequencies, the real portion of VI is constant, and the 
imaginary portion proportional to /, so that the resulting im- 
pedances as given by (12) are periodic in respect to the frequency, 

with a period of /o = . — = / , which is the number of 

V LCI vLtCt 
cycles increase in frequency required to bring about a recurrence 
of the same values of impedance. 

It will be remembered that, in the case of a line with no losses, 
a = 0, the impedances were both periodic in I with / constant, as 
well as periodic in / with I constant; and further, periodic in the 
product fl. The impedance of a line with losses is periodic in / 
at infinite frequencies only, while with fixed frequency it is never 
strictly periodic in length as independent variable. If the line 
length be increased, both components of VI increase proportionately, 
so that, though the resulting curves showing the equivalent resist- 
ance and equivalent reactance, when plotted in terms of line 
length, present some of the characteristics of a periodic function, 
they are not strictly periodic since they do not present recurrently 
the same sequence of numerical values. 

98. Infinite-line Impedance. — If the line be infinitely long, 
then VI becomes infinity, and since both the hyperbolic tangent 
and cotangent approach unity for infinite values of argument 
(the argument may be either real or complex), the values of open- 
circuit and short-circuit impedance approach each other for 
increasing line length, and finally become 



"-VI - 



infinite-line impedance. (13) 



If the attenuation constant a be zero, in which case V is a pure 
imaginary, the impedances do not approach U in value with in- 
creasing line length; for, in such an event, the hyperbolic tangent 
and cotangent reduce to the trigonometric tangent and cotangent 



162 



THE EQUIVALENT IMPEDANCE OF CIRCUITS 



respectively (neglecting multiplication by ±.j), so that for I = oo 
the expressions for impedance become indeterminate. In no 
physical line is it possible for a to be absolutely zero, so that the 
impedances always approach U = infinite-line impedance, as a 
limit. 

Equation (2), the general expression for equivalent impedance 
when the line is neither short-circuited nor open-circuited, also 
reduces to U for infinite values of I — that is, the equivalent 
generator-end impedance in such a case is independent of the load- 
end impedance, Z . 

99. Numerical Illustrations of Equivalent Line Impedances. — 
For the power transmission line, whose constants are given in 
Tables XII and XIII, the following numerical values of open- 
circuited and short-circuited impedances are obtained, by equations 
(4) and (5) for a line length of 100 miles. 



TABLE XXI 

Short-circuit and Open-circuit Impedances op a 100 Mile Line 
at Ddtferent Frequencies 



r = 0.275, L = 0.00204, 


y = 0.15X10-», C = 0.0146 X10- 8 


/ 


Impedance in ohms 


Short-circuited 


Open-circuited 





27.5 +j0 




66,680.0 -jO 


25 


27.6+J32.1 




293.2 -j'4331.1 


60 


28.3+J77.8 




58.7 -j 1789. 8 


180 


36.8+J264.5 




15.2-;'526.7 


300 


75.8+J616.4 




12.7-j'225.2 


420 


865.7+;' 2593. 9 




13.9 -j49. 2 


540 


165.3 -j 1272.0 




18.0 +j 107.5 


660 


29.5-J449.4 




29.6+J309.0 


780 


16.2-J188.2 




82.0+ j 735.4 


900 


14.6 -j20. 8 




3,255.5 +./4428. 1 



The curves in Figs. 59 and 60 are plotted from the above 
values. 

In this case, where the line losses are very small, the equivalent 
resistance and reactance reach enormous values. For this line, 
quarter-wave-length phenomena appear at a frequency of 458.06 
cycles per second, and half-wave-length phenomena at 916.12 cycles 
per second. In the short-circuited condition, the impedance 



NUMERICAL ILLUSTRATIONS OF LINE IMPEDANCES 163 





















rtj 


\ 
1 
















[600 


r 

m) 




















































71 


















L200 


000 
















/ 




































/ 


I 
















800 


too 
















' f 




































rl 


















400 


•m 
















I 


V 


























x^ 










































pf 


■qlt 


anov. f 


a; 














=-- 









K 





a 


10 


8 


K) 


400 


500 


a 





7f 


>0 


800^- 




400 
























































| 






•^CC 










800 
























































' 
















1200 
























































x, 1 
















Fig. 59. — The Equivalent Impedance of a Short-circuited, 100 Mile 

Power Line. 


X 


r 




























'7 






700 


70 


































600 


GO 


































500 


50 




V 






























400 


40 


























/' 








300 


:so 


































200 


20 


































100 


10 
















r 




^ 

















n 










Pre 


iue 


icy 


/ 


















-100 







1( 





a 


X) 


a 


Q 


400/ 


1 500 


8 


X) 


K 


a 


8 


X) 


900 


-200 




































-300 














/a 






















-400 




































•500 




































-600 


































-700 





































Fig. 



60. — The Equivalent Impedance of an Open-circuited, 100 Mile 
Power Line. 



164 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

reaches very large values at the frequency which makes the line a 
quarter-wave length. Beginning at zero frequency, the reactance 
increases approximately along the tangent curve shown in Fig. 57, 
but, instead of reversing sign through infinity, reaches a finite 
maximum value at a frequency slightly less than 458 cycles, drops 
rapidly through zero and to a negative maximum at a frequency 
slightly greater than 458 cycles. The numerical values of this 
negative, or condensive, reactance decrease, then, along a curve 
approximating the tangent curve, until at a frequency of 916 
cycles the half-wave-length phenomena appear and the reactance 
passes through zero. With continuously increasing frequency, 
above this value, the curves are sensibly periodic, for, from Table 
XIX, a has become practically constant. It is interesting to note 
that, at the frequency corresponding to half-wave-length phenom- 
ena, the effective generator-end resistance is only a little more 
than one-half its zero-frequency value. 

The open-circuited impedance is, of course, very high at low 
frequencies. The equivalent resistance at zero frequency is 66,680 
ohms — slightly greater than the reciprocal of the total line 
conductance, 

10 6 
100 X 0.150 = 66>6670hms - 

The equivalent reactance at zero frequency is zero, but it rises 
rapidly to very high numerical values, approximating the recip- 
rocal of the total line susceptance. The numerical value of this 
negative, condensive, reactance decreases along a curve approxi- 
mating the cotangent curve shown in Fig. 58, passes through zero 
at a frequency of 458 cycles per second, where the quarter-wave- 
length phenomena appear, and then continues approximately 
periodic in frequency as independent variable. At a frequency of 
300 cycles, the equivalent resistance is less than one-half the total 
line resistance. 

When longer lines or lines with larger values of r and g are 
considered, the equivalent resistance and reactance do not pass 
through such extended ranges of values. As a comparison with 
the above case, the values of the short-circuited and open-circuited 
impedances of the No. 12 N. B. S. gauge telephone circuit, for 
which the values of a and /3 are given in Table XX, are tabulated. 
The length of line is taken as 300 miles. 



DETERMINATION OF LINE CONSTANTS 



165 



TABLE XXII 

Short-circuited and Open-circuited Impedances 
of 300 Mile Telephone Line 

r = 9.94, L = 0.00366, g = 1.5X10-*, 
C = 0.00822X10^ 



/ 


Impedance 


Short-circuited 


Open-circuited 




5 

10 

20 

30 

50 
100 
150 
200 
300 

400 
500 
600 
700 
800 


2112 -yo 

2108 -y 80 
2095 -j 158 
2047 -y 307 
1973 -y 437 

1776 -y 627 

1291 -j'745 

1003 -y 642 

865 -y 514 

791 -y 345 

773 -y 297 
720 -y 263 
698 -y 200 
713 -; 177 
693 -y 176 


3137 -yo 

3074 -; 374 
2902 -y 689 
2423 -y 1047 
1991 -y 1125 

1478 -y 989 

1093 -j 651 

995 -y 513 

936 -y 461 

808 -y 397 

732 -j 297 
732 -y 230 
722 -y 221 
686 -; 189 
692 -y 148 



As may be seen by an inspection of the values of impedance in 
Table XXII, a line of this length and with these values for the 
fundamental constants, has no inductive reactance at low fre- 
quencies even when short-circuited. For the range of frequencies 
covered in Figs. 61 and 62, plotted from the above table, the 
reactance is condensive throughout — both when short-circuited 
and when open-circuited. At very high frequencies, the equiva- 
lent reactances decrease, numerically, and finally oscillate about 
a zero value, while the equivalent resistances oscillate about some 
finite limiting value. 

A comparison of Figs. 61 and 62 with Figs. 59 and 60 shows 
strikingly the effect of an increase in the values of line resistance 
and leakage, and therewith an increase in the attenuation con- 
stant a. 

100. Determination of Line Constants from Short-circuit and 
Open-circuit Tests. — It is frequently necessary to determine, 
by actual measurement, the effective values of the four funda- 



166 



THE EQUIVALENT IMPEDANCE OF CIRCUITS 





m 


r 






















































































m 


MOO 


/a> 






























































The eft e 


•tivc 
v.- tl 


MM MM in 










rm 


iOOO 


\ 1 




















rni 


&■ " 


uencica ihowo 




















































p 


100 


po 
















































V 






































BOO 


200 
















^r 






































































200 


800 






























































;• 






















_ 


100 


(00 






































































































Fre 


|UC 


ncy 


ft 


\Vul 


J 1 


itS 


I't'O 


ad 
















f 


100 

1 


i 





300 


400 


, 





" 





T 


800 







Fig. 61. — The Equivalent Impedance of a Short-circuited, 300 Mile 
Telephone Line. 



X 

■1120 


r 

2S0O 




































^ 




■1040 










































••960 


21011 
























I 
COI 


le effective i 
idenslve|thrc 


'eactaDCt 

uriiuut 


13 

lie 




•■880 


























ran 


geo 


I In 


que 


icie 


-Shr 


VTI1. 




■800 


20(H) 




V' 




































•720 










































••640 


1600 








































•560 










































■•480 


1200 








































•■400 






A 




































■320 


800 
















aj 
























•240 
































r 


























-160 


400 




























S**" 












-80 


























































R 


en 


enc 


y.f 


Cyc 


M 1 


>erf 


C'CO 


n<l. 








m 








) 


too 


2 


• 


300 


400 


500 


600 


71 


') 


i 


ID 







Fig. 



62. — The Equivalent Impedance of an Open-circuited, 300 Mile 
Telephone Line. 



DETERMINATION OF LINE CONSTANTS 167 

mental line constants at some particular frequency. On account 
of the variation of these constants with changes of frequency, it is, 
in general, not possible to measure the resistance and leakage of 
the line with direct current, nor to measure the capacity and self- 
mductance by any ballistic galvanometer method. If alternating 
current be used to make such determinations, all of the line con- 
stants come into play, so that any one measurement will not serve 
to determine the four constants. The frequency used in the 
measurement must be the same as that for which the values of r, 
L, g, and C are desired. Of course, in some cases, where approxi- 
mate results only are desired, such simple procedures as the deter- 
mination of the line capacity from an open-circuit test by the 
equation 

Ct = ttTW farads, 

where E g and I g are respectively the generator voltage and gen- 
erator current when the line is open-circuited, may suffice. How- 
ever, if the line is electrically long — that is, approaches a quarter 
wave-length — or if the resistance or leakage coefficients are large, 
such a procedure may not yield even an approximate result. 
Accurate values of the constants may be obtained by using two 
measurements — one of the short-circuit impedance and one of 
the open-circuit impedance. When both components of the imped- 
ances are measured, four numerical values are obtained as data 
from which the values of the four line constants may be calculated. 
Suppose that, by measurement at the particular frequency for 
which the effective values of the fine constants are desired, actual 
numerical values are obtained for 



and 



Zg = short-circuit impedance 
Z g " = open-circuit impedance, 



the line length, I, being known. The unknown quantities are 
Z and Y per unit length, or U and V, which are expressed in terms 
of Z and Y by equations (4) and (6), Chapter VI, and from which 
Z and Y may be obtained. 
Now 

Z ' = UttmhVl 
and 

Z " = UcothVl, 



168 THE EQUIVALENT IMPEDANCE OF CIRCUITS 



so that by multiplication 

U = VZ g 'Z g ". 

Further, 

sinh VI e w - <-n _ 1 - t~™ 

so that 

l- € -2Vl 



(14) 



tanh VI = 



cosh VI 
Z.'» I 



1 + e 



,-2 VI' 



(15) 



which, on solving for e 2 n = ryf|i gi yes 



2Vl _ U + Z g ' 
~U-Z a '' 



or 



v = h ln zr^6 -nw+m. 



(16) 



where the notation, In x, signifies "the logarithm to base c of x." 

U 4- Z ' 
The quantity 77 -^-, is complex, of the form A -\- jB f and it 

U Zlg 

is of this quantity that the logarithm is required. Since 

e «*+>" = c « (cos v +j sin v) = e u /v (radians) , 
we have 

u-\-jv = In e u (cos v + j sin v) = In (e u [v). 

Therefore, 

In (A + jB) = ln(VA 2 + B 2 /4) 
= \nVA 2 + B 2 +jd, 



where 
Thus 



B 

6 = arc tan -r- 
A 



V = ~ln VA 2 + B* +j ± arc tan | 
With C/ and V from the preceding equations, 

U = sJy and V = VzF, 



Z = r+jaL = UV and 
which thus determines the four line constants. 



(17) 



(18) 



Y = g+ju>C = $j, (19) 



METHODS OF MAKING IMPEDANCE MEASUREMENTS 169 

101. Multiple Solutions. — In using equation (18) it is im- 
possible to avoid a certain ambiguity. In taking the arc tan -j, 

r A 

any multiple of 2 ir may be added to the result at will, thus per- 
mitting of a variety of values for the imaginary portion of V. 
Multiplying (18) by Z, 

Vl = al+ j(3l = ^ In VA 2 + B 2 + j i arc tan j- 



al = 1 In VA 2 + B 2 , 
(31 = ■= arc tan -j- 



(20) 



When the line length and frequency are not too great and the 
conductors are in air, an estimate as to the probable value of /3Z 
may be made in order to decide whether or not a multiple of 2 w is 

to be added to arc tan -j. In (20), 01, itself, may differ from the 

true value by any multiple of ir, since one-half the arc tan -j is 

involved. If the measurement be made upon a circuit consisting 
of a long cable, where it would be impossible to estimate accurately 
the number of wave lengths involved, it would be necessary to 
commence the measurements at frequencies for which the cable 
would certainly be less than one-half wave length, and then, by 
taking measurements at a number of increased frequencies, to plot 
the impedance curves in order to decide upon the number of times 

ir must be added to ^ arc tan -j in determining &l by (20). The 

quantity a is obtained without any difficulties of this kind. When 
the line is very long, the differences between the measured imped- 
ances and U will be very small, since U is the infinite-line im- 
pedance, so that considerable inaccuracy may result from equation 
(16) on account of the small difference between two nearly equal 
quantities in the denominator. 

102. Methods of Making Impedance Measurements. — The 
impedance measurements may be made by the voltmeter-ammeter- 
wattmeter method, or by means of an alternating current bridge, 
depending upon the amount of power available and upon the 
character of the transmission line under investigation. When the 



170 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

first method is used, in the case of a power line with large con- 
ductors, it will probably be difficult to obtain the real component 
of the impedance accurately on account of the very low power- 
factors — particularly with the line open-circuited. When an alter- 
nating current bridge is used, the current and voltage impressed 
upon the line will probably be much smaller in value than those 
obtaining when the line is in actual use. However, as long as no 
corona formation occurs when the line is in use for its regular 
purpose, and no ironclad circuits are present, the values of the 
constants obtained by a bridge method with small amounts of 
power would not be subject to change when greater voltage and 
current are used. The voltmeter-ammeter-wattmeter method of 
measuring both components of impedances is so familiar as not to 
need description. 

103. An Impedance Bridge. — A form of alternating current 
bridge which the writer has found to be very satisfactory is shown 
in Fig. 63. 

In the two upper figures, a and b, the bridges are shown dia- 
grammatically, and in the lower figure are shown the connections 
to a double-throw, double-pole switch to be used for changing 
from scheme a to scheme b. R h R 2 , and R are adjustable non- 
inductive resistances, and C is an adjustable standard capacity. 
Connection "a" gives directly the two components of a condensive 
admittance, Y = g + juC, so that it is necessary to take the 
reciprocal of this value to obtain the impedance. Connection 
"b" gives the two components of an inductive impedance, Z — 
r + jb)L. The formula? to be used in the reductions are given 
beneath the diagrams, and in them the values of R, R h R 2 , and 
Co which produce a balanced condition in the bridge are to be 
substituted. To detect a condition of balance, either a telephone 
receiver or a vibration galvanometer may be used — the latter 
being, in many instances, the more satisfactory. When a telephone 
receiver is used, frequencies much below 250 cycles per second 
cannot be employed with the bridge on account of the inaudibility 
of their tone, and again, with any circuit as unknown whose 
apparent constants change with the frequency, a balance at one 
frequency will not be a balance at another frequency, so that with 
any ordinary source of e.m.f. difficulty will arise from the presence 
of higher harmonics in the e.m.f. wave. The sound produced by 
them will always be present and serve to mask the disappearance 



PRECAUTIONS IN THE USE OF THE BRIDGE 



171 



of the fundamental tone at a condition of balance. The vibration 
galvanometer, on the other hand, may be used at low frequencies 
as well as at frequencies of hundreds of cycles per second, and since 
the period of the instrument is adjusted to coincide with the 
period of the e.m.f. used, no trouble will arise from the presence of 
higher harmonics of ordinary magnitudes — say a few per cent. 




Connection a. 

" R Rj 

c=£c 



Connection 6. 
r _ R» R2 



L = R 



Co 



<\j 



C 



AA/V\AA 



K^> 



AA/WNA 




Unknown 



Fig. 63. — An Impedance Bridge. 

For Ri and R 2 , resistances with several steps and with a total 
value of about 2000 ohms each are appropriate, except for the 
measurement of small inductances, by connection "b," where 
values as low as 50 or 100 ohms may profitably be employed. 

104. Precautions in the Use of the Bridge. — To avoid 
excessive values of R it is often desirable to shunt the unknown 
with a known non-inductive resistance when measuring condensive 
admittance by connection "a," or to insert a known non-inductive 
resistance in series with the unknown when measuring inductive 



172 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

reactance by connection " b," The values of g or r thus introduced 
are to be subtracted from the measurements. The alternating 
e.m.f. supplied to the bridge should be taken from the secondary 
of an air-core or other insulating transformer placed at a distance 
sufficient to avoid the influence of extraneous magnetic fields. 
This insulating transformer is needed to prevent the introduction 
of errors on account of the large electrostatic capacity between the 
generator windings and leads and earth. If great accuracy is 
desired, the different portions of the bridge should be electro- 
statically shielded, and, in any case, care should be taken so to 
distribute the different parts of the apparatus that the mutual 
interactions be reduced to a minimum. 

Considerable difficulty is often experienced in finding the proper 
adjustment for a balance. When measurements of a transmission 
line are undertaken, one usually has no knowledge of the approxi- 
mate values, even, of the unknowns, so that a balance must be 
sought by trial. When a long line is involved and the frequency 
is high, it cannot be known beforehand whether the unknown im- 
pedance is inductive or condensive. Trial must then be made of 
both bridge connections. Since there are two quantities which 
require simultaneous adjustment, C and R, the best that can be 
done is to first adjust one of them until the current in the gal- 
vanometer is a minimum, then adjust the other until a new mini- 
mum is obtained, and so on. A little experience enables these 
adjustments to be made very rapidly after a condition is found 
where a change in either C or R produces a noticeable change in 
the galvanometer current. Care must be taken to see that the 
frequency used does not approach those for which the impedances 
under measurement vary rapidly, for then small variations in the 
frequency of the e.m.f. supplied to the bridge would cause the 
bridge settings for a balance to vary greatly, and thus prevent any 
accurate adjustments. For example, in an impedance measure- 
ment of the short-circuited 100 mile line, whose effective resistance 
and reactance are shown as a function of the frequency in Fig. 59, 
reliable measurements would be difficult with frequencies between 
350 and 500 cycles per second. According to Fig. 60, for this same 
line open-circuited, measurements would be difficult with fre- 
quencies between 800 and 1000 cycles per second, as well as at 
450 cycles, where the impedance would change from inductive to 
condensive, or vice versa, for small changes of frequency. 



EFFECT OF ERRORS IN ORIGINAL DATA 



173 



105. Illustrative Computation. — To illustrate the use of 
equations (14) to (19), consider the following numerical example. 
On a given line, 41 miles in length, impedance measurements with 
the following results were made at a frequency of 660 cycles per 
second : 



Line short-circuited. Connection " b " 


Line open-circuited. Connection " a " 


ft = 1000 Ri = 1000 

R = 1224.0 Co = 0.2766X10"* 


Ri = 1000 R 2 = 1000 

R =816.7 Co = 0.7816X10-* 



By using the respective formulae in the two cases, "a" and "b": 
Short-circuit test, "b," 

Z ' = 817 +i 1147 ohms. 
Open-circuit test, "a," 

Y g " - 0.0012244 +j 0.0032411 mhos. 
1 



Y " 

1 o 



= Z g " = 102.0 - j 270.0 ohms. 



These values are to be used as original data with which to enter 
equation (14). The computations are most easily done with 
logarithms, according to the following table, where in place of the 
quantity itself, its logarithm is given, unless otherwise stated. 
<f> represents, in general, the angle of a complex quantity; for 

example, 

Z = r +jx = z[$. 

These are the effective values of the line constants at this fre- 
quency (660 cycles per second) per mile of length, and they will, 
in general, be different for different frequencies. 

106. Effect of Errors in Original Data. — In this numerical 
example, the fundamental data, Z g ' and Z g ", are such as to yield 
very accurate results, for the two impedances as measured are 
sufficiently different from each other to permit an accurate for- 
mation of their sums and differences, as required in equation (16). 
Further, the resulting values of impedance and admittance per 
unit length of line have angles which are not very nearly equal to 
either or 90 degrees. The determination of the quantity g is 
usually the most unsatisfactory, for the angle of the admittance, 
Y, is frequently very near 90 degrees, so that g, the real component, 



174 



THE EQUIVALENT IMPEDANCE OF CIRCUITS 



TABLE XXIII 

Calculation of Link Constants from Two Impedance Measure- 
ments. Equations 14 to 20 



EJ 




2.91222+ j 3.05956 


tan <t> 


0.14734 


sin <f>' 




9.91089 






Z " 




2.00860 -j 2. 43136 


tan <t>" 


-0.42276 


sin*" 




- 9.97103 






z„' 




3.14867/54.538 deg. 






*'" „ 




2. 46033/ -69. 305 deg. 






. 




5. 60900/ -14. 767 deg. 






COS * u 




9.99638-10 






V 




2. 80450/ -7. 3835 deg. 






sin 4> u 




-9.10893-10 






U 




2.80088 -j 1.91343 






u 




632.25 -J81.93 t Numericallv 
817.00 +j 1147.00 f Numerically 




*; 






Let m = 


U + Z ' 


' and n = U - Z„' 






m=U + Z„' 
n=U-Z a ' 


1449.25 +j 1065.07 ) M 
-184.75 -j 1228.93 J w 


umerically 




m 




3. 16114 +j 3. 02738 


tan <f>m 


9.86624-10 


COS <j>m 




9.90623-10 






n 




2.26659 -j 3. 08953 


tan <f> n 


-0.82294 


sin* n 




- 9.99515 







m 3.25491 /36.313 deg. 
n 3.09438 /261.450 deg. 



A +JB = - 0.1605 3/134.863 deg. 



(Logarithm of scalar) 



log log - 


9.20557-10 


<f>° = tan- 1 j 


134.863 deg. 


log Mod. 


9.63778-10 


log* 


2.12989 


log In — 
n 


9.56779-10 


, 360 

log 2T 


1.75812 


log 2 J 


1.91381 


log arc tan -r 


0.37177 






log 2 1 


1.91381 


s'm<t> v 


7.65398-10 +j 8. 45796-10 tan <f> v 0.80398 
9.99471-10 
V 8.46325-10/81.075 deg. 
U 2.80450 -7.383 deg. 


4>z 

cos 4> t 
z 

sin <t> z 

r 

X 


73.692 deg. 
9.44841-10 
1.26775 
9.98216-10 
0.71616 
1.24991 
3.61772 
7.63219-10 


4>v 

cos 4> v 

y 

sin 4> v 

I 
2 c f 


88.458 deg. 
8.42985-10 
5.65875-10 
9.99984-10 
4.08860-10 
5.65859-10 
3.61772 
2.04087-10 




Final numerical results 




r = 5.2018, 


L = 0.0042873 


, = 1.2263Xl(H 


C = 010987 X lO-* 



EFFECT OF ERRORS IN ORIGINAL DATA 175 

is not accurately determined. In the particular case just treated, 
the angle of Y is 88.458 degrees — only 1.542 degrees different 
from 90 degrees. In general, the error in the angle of Y is of a 
lower order of magnitude than the original errors in the measured 
angles of the equivalent impedances, Z g and Z g ", so that in this 
case, the cosine of the angle 88.458 degrees, on which the determi- 
nation of g depends, is still obtained with considerable accuracy. 
To illustrate the effects of an error in the original measurements, 
suppose that, in determining Z g ', an error of one degree is made 
by a wrong determination of the imaginary component of the 
impedance — the real component being assumed correct. Then 
as original data, in logarithms, 

Z g ' = 2.91222 + j 3.07570 = 3.15951 /55.538 deg. 
Z " = same as before. 

From this fundamental data, 

r = 5.0394, L = 0.0043677, g = 1.4026 X 10"*, 
C = 0.010874 X 10- 6 . 

In this latter case, the angle of Y is 88.219 degrees, as compared 
with 88.458 degrees in the previous case based on the correct 
values of Z ' and Z g '\ The change in the resulting angle of Y 
due to a change of one degree in the angle of Z g ' is 0.239 degrees. 
As mentioned before, the best results from measurements of 
this kind are obtained under conditions which cause the two 
impedances, Z g and Z e " } to be considerably different, for then the 
value of the fraction in equation (16) is obtained without loss of 
accuracy. The shorter the line under test, the greater the differ- 
ence in these impedances. With very short lines, however, when 
the resistance and leakage coefficients are small, it will be difficult 
to obtain accurate values of the real components of the effective 
impedances by measurement, so that, though the subsequent 
numerical solution will be easy to carry out accurately, the errors 
in the fundamental data may be serious. With the 300 mile tele- 
phone circuit, whose impedances are tabulated in Table XXII, 
for example, values of line constants based on impedance measure- 
ments at frequencies above 100 cycles would be quite inaccurate, 
on account of the very nearly equal values of Z g ' and Z g ". If the 
effective line constants were desired at higher frequencies, it would 



176 THE EQUIVALENT IMPEDANCE OF CIRCUITS 

be desirable, or even necessary, to make the impedance measure- 
ments on a shorter section of the line — say 100 or 50 miles, 
depending on the frequency. On the other hand, measurements 
on a 300 mile power transmission line, whose values of r and g are 
usually much smaller than for the telephone circuit, would prob- 
ably furnish very satisfactory data. 



CHAPTER IX 
CHANGE OF WAVE SHAPE ALONG LINES 

107. Method of Treatment of Complex Wave Shapes. — The 

equations given in the foregoing chapter are all based on the 
complex quantity representation of alternating quantities, and 
thus yield solutions for a single frequency. By the definition of 
an ideal line r however, in which it is stated that the effective values 
of the line constants in such an ideal line are independent of the 
magnitudes of the voltage or current, the treatment of complex 
wave shapes by the superposition of separate solutions for the 
separate frequencies is made possible. If the e.m.f. wave of a 
generator which supplies a particular line contains a number of 
higher harmonics, the voltage wave shape at any point in the line 
may be found by determining the magnitude and phase position 
of the several component voltages at the point in question by 
applying the complex quantity method of analysis to each sepa- 
rately, and then taking the instantaneous sums as data from which 
to plot the resultant curve — if the curve, and not merely the 
values of the separate components, be needed. The method of 
treatment is essentially the same as used in the correction of 
oscillographic curves of wave shape for the effect of distortion 
circuits, Chapter IV. If any of the generator e.m.f. harmonics 
should coincide in frequency with a resonance frequency of the 
line, and the line losses be small, very great exaggeration of this 
particular harmonic, either in the voltage or in the current pro- 
duced thereby, may take place in portions of the line. For a con- 
sideration of such phenomena, it will be necessary to reproduce, in the 
form for use in alternating current systems, some of the specialized 
equations given in Chapter II for leaky direct current lines. 

108. General Case ; Load Impedance = Z . — The funda- 
mental equations are: 

E = E- cosh VI + hU sinh VI, 



I = 7 cosh VI + E A sinh VI, 
177 



(1) 



178 



CHANGE OF WAVE SHAPE ALONG LINES 



which give the voltage and current at any point distant I from the 
load-end of the line in terms of the load-end voltage and current. 
Solving simultaneously for E and h (see eq. 16, Chap. II), 



E = E cosh VI - IUsinh VI, 
I = IcoshVl-EjjeinhVl, 



(2) 



which give the voltage and current at any point distant I from the 
generator in terms of the generator voltage and current, E and /. 
To determine the voltage and current at any point in a given 
line when the load-end impedance, Z , is known, let 

k = total line length between generator and load, Z , 

and let E' and /' represent the voltage and current respectively at 
any point distant V from the generator, whose voltage is E. V is 
thus always less than lo. 

The ratio between current and voltage at the generator is the 
equivalent generator-end impedance, which, by equation (2), 
Chapter VIII, is 

Z cosh Vk+U sinh VI* 



Z a = 



Y a = 



cosh Vlo + jj sinh V k 



1 



and / = EY a 



(3) 



Therefore, applying equation (2), for a distance I' from the genera- 
tor and eliminating I by equation (3), 

E' = E (cosh VV - UY sinh VI'), 



r ' = e( 



Y cosh VV -jj sinh 



')1 



(4) 



In computing from this equation, first determine the value of Y g 
and then substitute it in the equation in connection with the 
particular values of V. No benefit is to be derived from the 
substitution of the expression in (3) for Y g in equation (4), except 
numerically, as mentioned. 

109. Short-circuited and Open-circuited Line. — Since con- 
siderable labor is required to determine the values of Y„, in general, 
the two special cases — short-circuited line and open -circuited 
line — are" of importance, for then the values of Z g or Y g are easily 



SHORT-CIRCUITED AND OPEN-CIRCUITED LINE 



179 



obtained. For these special cases, the substitution of the expres- 
sions for the equivalent generator-end impedance from equations 
(4) and (5), Chapter VIII, gives 

Short-circuited : 

sinh V(k-l') 



E'~\ = E (cosh VI' - coth Vk sinh VI') = E 
Jz„=o 

/'] = 5 (coth Vk cosh VI' - sinh VI') = E 
Jz =o U 



sinh Vlo 

coshV(lo-l') 

U sinh Vk 



(5) 



Open-circuited : 



E'~\ = E (cosh VI' - tanh Vl sinh VI') = E 

Jz =oo 

V\ = ^ (tanh Vk cosh 7Z' - sinh VI') = E- 

_|Z„=oo C 



coshF(Zo-Q 

cosh F/o 

sinhF(Z -ZQ 

[/ cosh Vlo 



(6) 



If we make V = Zo, then equation (4) must reduce to forms 
similar to equations (19) and (20), Chapter II — giving the load 
voltage and current in terms of the load impedance, generator 
voltage, and total line length. 

Making the substitution, V = k, and reducing, 

F % 

cosh Vk +Y Usmh Vlo' 

r g 

Zo cosh Vk + U sinh Vk 

From these, for the two special cases, 

Eo\ =EsechVk, 

JZ =<x 

I 



(7) 



r = E -p: cosech Vk- 

Jz =o U 



(8) 



In a direct current system, the quantity V is always real, and 
thus in equation (8), for instance, the quantity E sech Vk con- 
tinuously decreases for increasing line length. In the alternating 
current system, V is complex, and a function of the frequency, so 
that the value of sech Vk may vary through very great numerical 
ranges, and in the case of a line with no losses in which the attenua- 
tion constant is zero, throughout the range from — oo to +«>. 
Similarly with cosech Vl - The voltages at the terminal of an 
open-circuited line may thus rise to very high values, as well as 



180 CHANGE OF WAVE SHAPE ALONG LINES 

the current at the end of a short-circuited line. Equations (5) 
and (6) indicate that excessive voltages or currents may also occur 
at intermediate points in the line length, if the total line length, 
k, be such as to give large values to coth Vk or to tanh Vk, depend- 
ing upon whether the line is short-circuited or open-circuited. If 
the line be connected to an impedance Z , the excessive currents or 
voltages may or may not occur, depending on the resulting value 
of Y g , from equation (3). Y g becomes very large when 

Z cosh Vlo+ U sinh Vk approaches zero. 



For example, 


in 


a line with no 

cosh Vk 
sinh Vk 

U 


losses (a = 

= cosjSZo, 
= j sin Pk, 


0), 


so that 











Y g = oo when Z cos fik + j y -x sin &k = 0, 



or when 



'o = -JVc tan ^- 



Infinite values of Y g , however, imply an equivalent short- 
circuit on the generator supplying the line, and under such a 
condition it is not reasonable to suppose the generator voltage to 
remain constant, for then we would have infinite currents through- 
out the line, as well as infinite voltages. In such cases — very 
large values of Y g — the phenomena must be expressed in terms of 
the generator current J, rather than the generator voltage, E, 
the former being more or less definite at short-circuit, while the 
latter disappears. The equations in terms of I are obtained by 
substituting IZ g for E, and then, from equations (4), (5), (6), (7), 
and (8), we have 

E' = I (Z g cosh VV - U sinh VV), 



r = I (cosh W - jf sinh Vl'\ 

E'\ _ = IU(tsLBhVkcoshVV-8ii^Vn=IU sm ^y~ V) , 

/'] = I (cosh W - tanh Vk sinh VV) = I coshy (*»-*'> 
Jz =o coshyfo 



(9) 



(10) 



NUMERICAL ILLUSTRATION 



181 



coshF(Z -O 



1 

W =IU(cothVl Q coshVl'-smhVl')=IU 

/'I = I (cosh W - coth Vk sinh VV) - 7 - smh 7 ( *° ~ 

-^o = i 



sinh VL 



(11) 



Y cosh F^o + yy sinh Vl 



Io = 



cosh VIq + yr sinh FZ 
o = IU cosech FZo, 

JZ =oo 

o = 7 sech FZo- 



(12) 



(13) 



These equations will serve to determine the voltage at any point 
in a line, under any condition, and will also furnish the basis for 
a discussion of the phenomena of quarter-wave transmisson. 

Consider first the distortion of wave shape of voltage in an open- 
circuited line when the generator voltage contains higher har- 
monics of small magnitude in comparison with the fundamental. 
The wave shape of the generator voltage then differs little from a 
sine curve. Of course, if no higher harmonics are present in the 
original wave shape of the generator, there will be no distortion. 
Referring to equation (6), suppose the line length to be one- 
seventh of a quarter wave length for the fundamental, or approxi- 
mately, j3i°Zo = 13 degrees. When the line losses are small, the 
voltage of fundamental frequency does not change very much 
throughout the entire line length. The third and fifth harmonics 
of voltage are considerably magnified, while for the seventh har- 
monic, tanh Vk becomes very large, infinity if a be zero, so that 
this particular harmonic attains values relatively much greater 
than either the third or the fifth. It will thus greatly alter the 
shape of the e.m.f. wave at points along the line, as well as at 
the load end. A numerical example serves best to illustrate this 
effect. 

110. Numerical Illustration. 100 Mile, Open-circuited Line. — 
The 100 mile transmission line whose fundamental constants are 

r = 0.275, L = 0.00204, g = 0.15 X 10" 6 , and C = 0.0146 X 10" 6 , 



182 



CHANGE OF WAVE SHAPE ALONG LINES 



has, for the frequencies which are the odd multiples of 60 cycles 
per second, and which are the only ones which would generally be 
present in the e.ra.f. wave of an alternator, the propagation- 
constants and open- and short-circuit impedances given in Tables 
XIX and XXI. In this case lo = 100 miles. 

The coefficients of E in equations (6) may be termed the voltage 
and current coefficients, respectively. Numerical values of these 
coefficients for different frequencies and different distances V, 
from the generator are given below in Table XXIV. 



TABLE XXIV 

Factors for Obtaining the Distortion of Voltage and Current 
Wave Shapes along a 100 Mile Line when Open-circuited 

Or 







(a) Voltage coefficients, -=■ , as 


scalar with angle 






/ 


/' = 25 


i' = 50 


J' = 75 


1 


' = 100 


60 


1.009 


- 0°.21 


1.016 - 0°.36 


1.020 - 0°.44 


1.021 


- 0°.47 


180 


1.097 


- .76 


1.168 - 1 .25 


1.211 - 1 .52 


1.226 


- 1 .61 


300 


1.387 


- 2 .11 


1.684 - 3. 12 


1.871 - 3 .61 


1.934 


- 3 .76 


420 


3.476 


- 13 .63 


5.539 - 15 .81 


6.895 - 16 .60 


7.367 


- 16 .81 


540 


0.654 


-163 .03 


2.145 -170 .69 


3.192 -171 .92 


3.567 


-172 .20 


660 


0.203 


- 10 .38 


0.665 -174 .86 


1.319 -176 .91 


1.563 


-177 .27 


780 


0.473 


- 2 .51 


0.260 -174 .10 


0.878 -178 .41 


1.119 


-178 .86 


900 


0.678 


- 1 .72 


0.340 -144 .28 


0.717 -179 .32 


1.001 


-179 .87 





(6) Current coefficients, -= X 10* 
JS 


, as scalar with angle 






f 


r = o 


r = 25 


i' = 50 


V 


= 75 


60 


0.558 88°. 12 


0.420 88°. 05 


0.281 88°. 00 


0.140 


87°. 98 


180 


1.898 88 .35 


1.464 88 .14 


0.996 87 .99 


0.504 


87 .90 


300 


4.434 86 .77 


3.608 86 .38 


2.546 86 .12 


1.316 


85 .98 


420 


19.545 74 .24 


17.385 73 .63 


12.998 73 .25 


6.944 


73 .04 


540 


9.173 -80 .50 


9.387 -81 .46 


7.627 -82 .00 


4.262 


-82 .29 


660 


3.222 -84 .53 


4.149 -86 .19 


3.785 -86 .94 


2.242 


-87 .30 


780 


1.351 -83 .64 


2.714 -87 .34 


2.912 -88 .40 


1.855 


-88 .85 


900 


0.182 -53 .68 


1.971 -87 .68 


2.676 -89 .28 


1.867 


-89 .83 



Suppose that the generator supplying this line has an e.m.f. 
wave shape whose equation is, as a function of 6 = 2 irft, 

e = 100,000 sin d 

+ 8000 sin (3 0-40°) 

+ 6000 sin (5 0-295°) 

+ 5000 sin (7 0- 110°) 

+ 2000 sin (9 0-50°) 

+ 500 sin (11 0-170°). (14) 



EQUATIONS FOR INSTANTANEOUS VOLTAGES 183 

The equation for any particular component of the e.m.f. at any 
point in the line is obtained by multiplying the equation for the 
component of the same frequency in e, by the factor, or voltage 
coefficient in part (a) of Table XXIV, and then changing the 
phase angle as given in (14) by the angle of the factor. Thus, at 
a distance of 75 miles from the generator, the maximum value of 
the ninth harmonic in the voltage wave is 3.192 X 2000 volts = 
6384 volts, and its equation as a function of time (or 0) is 



e 9 | = 6384 sin (9 - 221°.92). 

J/'=75 



111. Equations for Instantaneous Voltages. — The following 
are the equations for the voltages at distances of 25, 50, 75, and 
100 miles from the generator. 

25 miles from the generator, V = 25, 

e' = 100,900 sin (0 - 0°.21) 

+8776 sin (3 - 40°.76) 

+8322 sin (5 0-297°. 11) 

+ 17,380 sin (7 0- 123°.63) 

+ 1308 sin (90- 213°.03) 

+ 102 sin (11 - 180°.38). (15) 

50 miles from the generator, V = 50, 

e' = lOl,6OOsin(0-O°.36) 

+9344sin(3 0-41°.25) 

+ 10,104 sin (5 0-298°. 12) 

+27,700 sin (70- 125°.81) 

+4290 sin (9 - 220°.69) 

+332 sin (11 - 344°.86). (16) 

75 miles from the generator, V = 75, 

e' = 102,000 sin (0 - 0°.44)' 

+9688sin(3 0-41°.52) 

+ 11,226 sin (50 - 298°.61) 

+34,480 sin (70- 126.°60) 

+6384sin(90-221.°92) 

+660 sin (11 - 346°.91). (17) 



184 CHANGE OF WAVE SHAPE ALONG LINES 

100 miles from the generator, V = 100, at open end, 
e' = lO2,lOOsin(0-O°.47) 

+98O8sin(30-41°.61) 

+ ll,6O4sin(50-298°.76) 

+36,835 sin (70 - 126°.81) 

+7134 sin (9 - 222°.20) 

+782 sin (11 - 347°.27). (18) 

112. Equations for Instantaneous Currents. — Using the 
current coefficients from part (6), Table XXIV, the following 
equations are obtained for the line currents at the several different 
distances from the generator. Obviously, the current is zero at 
the open end of the line, where V = Iq = 100. 

Generator current, at V — 0, 

t = 55,8OOsin(0 + 88°.12) 

+ 15,184 sin (3 + 48°.35) 

+26,604 sin (5 - 208°.23) 

+97,725 sin (7 - 35°.76) 

+ 18,346 sin (90- 130°.50) 

+ 1611 sin (11 - 254°.53). - (19) 

25 miles from the generator, V = 25, 

i' =42,000 sin (0 + 88°.05) 

+ 11,712 sin (3 + 48°.14)' 

+21,648 sin (5 - 208°.62) 

+86,925 sin (7 - 36°.37) 

+ 18,774 sin (90- 131°.46) 

+2074 sin (11 - 256°.19). (20) 

50 miles from the generator, V = 50, 

i' = 28,100 sin (0 + 88°.00) 

+7968 sin (3 + 47°.99) 

+ 15,276 sin (5 - 208°.88) 

+64,990 sin (7 - 36°.75) 

+ 15,254 sin (90 - 132°.00) 

+ 1892 sin (11 - 256°.94). (21) 

75 miles from the generator, V = 75, 

t' = 14,060 sin (0 + 87°.98) 

+4032 sin (3 + 47°.90) 

+7896sin(50-2O9.°O2) 

+34,720 sin (7 - 36°.96) 

+8524 sin (90- 132°.29) 

+ 1 121 sin (1 1 - 257°.30) . (22) 



DISCUSSIONS OF WAVE SHAPES 



185 




Fig. 64. — Voltage Wave Shapes along a 100 Mile Open-circuited 
Power Line. 



113. Discussions of Wave Shapes. — The curves representing 
the foregoing equations are shown in Figs. 64 and 64a. In both 
figures, the upper curve represents the generator voltage wave 
shape. These curves were drawn by means of the curve-tracer 



ISO 



CHANGE OF WAVE SHAPE ALONG LINES 




Generator Voltage 
Equation 11 



Generator Current 
Equation It 



Current at i •»» 
Equation 20 



Current at J'— 50 

Equation 21 



Current at I '=75 
Equation 22 



Fig. 64a. — Current Wave Shapes along a 100 Mile Open-circuited 
Power Line. 



mentioned in section 42, Chapter III, and, though they may 
be slightly inaccurate, serve sufficiently well to show the char- 
acteristics of the different wave shapes. 

In the particular line to which the foregoing curves apply, the 



DISCUSSIONS OF WAVE SHAPES 187 

total length of circuit is 100 miles, and therefore approximately 
equal to one-quarter of the wave length for the seventh harmonic 
— 109 miles, from the value of (3° given in Table XIX. The 
voltage and current coefficients for this frequency attain values 
much larger than for any of ( the other frequencies, and therefore 
the comparatively small seventh harmonic in the generator 
voltage produces very prominent voltages and currents of this 
frequency throughout the line. Although the seventh harmonic 
in the generator voltage is only five per cent of the funda- 
mental, the seventh harmonic in the voltage at the free end 
of the line is thirty-six per cent of the fundamental. In 
the capacity, or charging, current of the line, the seventh 
harmonic component is approximately twice as great as the 
fundamental. 

The maximum instantaneous values of voltage at different 
points along the line are indicated on the respective curves. On 
account of the different phase positions of the several component 
harmonics, the maximum value of the generator voltage wave is 
only a little greater than the maximum value of the fundamental — 
104,000 maximum, with 100,000 volts as maximum value of the 
fundamental. At the open end of the line, the maximum value 
of the voltage wave is 148,000 volts, which, as an increase of nearly 
fifty per cent over the generator voltage, might be serious in its 
effect upon the line insulation. In a line designed to operate 
normally without corona formation at the voltage assigned to the 
fundamental frequency in this illustrative example, the presence 
of such a large superimposed harmonic which increases the maxi- 
mum voltage by nearly fifty per cent might be sufficient to start 
corona formation, and introduce a power loss then supplied, in 
part, by the fundamental frequency. Very considerable values 
of leakage, or corona loss, are required to materially decrease the 
voltage coefficients and thus to decrease the exaggeration of those 
harmonics for which the line approximates a quarter wave length. 
The value of g used in the computation of the data for this particu- 
lar case is 0.15 X 10 -6 , which corresponds to a power loss due to 
leakage, of 1500 watts per mile of each wire for an e.m.f. of 100,000 
volts between wire and neutral. For a line differing, as in this 
case, by 9 miles from a quarter wave length for the seventh 
harmonic, this value of g is not sufficient to appreciably lower the 
value of the voltage coefficient in Table XXIV. 



188 CHANGE OF WAVE SHAPE ALONG LINES 

114. Dependence of Wave Shapes upon Generator and Load 
Characteristics. — The large value of the seventh harmonic in 
the generator current would seriously increase the generator 
heating, and demands, even under conditions of no-load, con- 
siderable generator capacity. The seventh harmonic current at 
the generator is 74.2 degrees ahead of the corresponding component 
in the terminal voltage, so that, for a considerable range of current 
values, whatever current of this frequency flowed through the 
inductive impedance of the generator windings would produce an 
increase of the terminal voltage (of this frequency) above the open- 
circuit value. No reliance could be placed on an assumption that 
in such a case the generator terminal voltage of this frequency 
would decrease greatly when the current flows. For a line a little 
longer than a quarter wave length, the current would lag by nearly 
90 degrees, though the numerical value of the voltage coefficient 
might be the same as before. In such a case, the terminal voltage 
of the generator undoubtedly would decrease materially as soon 
as the current flowed. 

An increase of only nine per cent in the frequency (from 60 to 65 
cycles for the fundamental) would cause the line to be an exact 
quarter wave length for the seventh harmonic, in which event, 
much greater distortion of the wave shapes would occur. A still 
further increase of the frequency would cause the line length to be 
greater than a quarter wave length for the seventh harmonic, and, 
depending upon the character of the source of e.m.f. of this fre- 
quency, might entirely alter the phenomena. 

The presence of a load at the terminal of such a line would also 
change the phenomena. If the load consisted of an inert imped- 
ance, Z , the distribution of electrical quantities along the line 
could be determined by equations (4), although not without 
rather tedious computations. If the load consists of a synchro- 
nous motor, for example, instead of an inert impedance, it is neces- 
sary to know the wave shape of counter electromotive force of this 
machine, and furthermore, to know the relative phase positions of 
its component harmonics of e.m.f. with respect to those of the 
generator supplying the line. The counter e.m.f. of fundamental 
frequency of the motor will be nearly in opposition to the e.m.f. 
impressed on the motor terminals, E , but, by reference to Fig. 33 
it is seen that the phase angle between the generator terminal 
voltage, E, and the load voltage, E Q , may differ by large amounts, 



ENERGY SUPPLY THROUGH TERMINAL IMPEDANCES 189 

depending upon the magnitude of the current I Q as well as upon 
the load power-factor. The possibility of a variation of, say, 60 
degrees in the phase difference between E and E 0) allows the 
higher harmonics in the counter e.m.f. of the motor to occupy 
almost any phase position whatsoever with respect to the same 
harmonic frequencies in the generator e.m.f. 

If a particular harmonic is present in the generator wave shape, 
but is not present in the counter e.m.f. of the motor, then the 
motor constitutes practically a short-circuit at the load end of the 
line for this particular frequency. Similarly, the generator con- 
stitutes a short-circuit at its end for any harmonic present in the 
motor counter e.m.f. which is not present in the generator wave. 
The discussion of the mode of operation of the various machines 
which may be connected to a line must be taken up in another 
place; this section is intended to deal only with the calculation 
of the values of the voltage and current along the line when the 
terminal conditions are given. 

If both the generator and the load produce e.m.f. 's, as in the 
case of a higher harmonic in both the generator and motor ter- 
minal e.m.f. 's, the distribution of voltage and current along the 
line must be determined by equations identical with (55) and (56), 
or (57) and (58), of Chapter II; in which, to conform to the nota- 
tion used here for alternating current systems, the complex 



Zi z 2 

Ei Ei Line Length = l e'„ 



Ii Zj 

Fig. 65. — Long Line Supplied through Impedances by Two Generators. 

quantities, U and V, must be substituted for z and v, respectively. 
Ei and E 2 are then the values (in the correct complex form to take 
account of their relative phase positions) of the voltages impressed 
on the two ends of the line, and E and I' and /" are respectively 
the values of voltage and current at a distance li from Ei and k 
from Ei. 

115. Double Source of Energy Supply Through Terminal 
Impedances. — If impedances are placed between the sources of 
e.m.f. and the line, the circuit appears as in Fig. 65, and the solu- 



190 



CHANGE OF WAVE SHAPE ALONG LINES 



tion must be obtained as below. In this system, select arbitrarily 
as positive the direction indicated by the arrow. Then, by 
Kirchhoff's Law, 

Ei = E\ — IiZ\, 

hiz = ivji — E%. 
By equations (2), 



(23) 



E t ' = Ex' cosh Vk - Uh sinh Vk, 
h = h cosh Vk-jj Et' sinh Vk, 



(24) 



or, substituting (23) in (24) and combining terms, 

7i [Zi cosh Vk + U sinh Vk] + ZJ* = E x cosh Vk + E 2 , 
1 1 cosh Vk + jj sinh Vk — h = Ex jj sinh Vk, 



(25) 



from which, by solving simultaneously for h and h, 



Ik-. 



/* = 



\-^8intiVk\ + E t 



(Zi + Z 2 ) cosh Vk + (u + ^A sinh Wo 

Ei + # 2 (cosh Vk + ^ sinh 7Zo) 
(Zx + Z 2 ) cosh FZo + (u + ^f- 2 ) sinh Wo 



(26) 



Using the value of I\ from equations (26), the voltage and current 
at any point in the line are then given by 



E = Ei' cosh VI - hU sinh VI, 
I = h cosh VI - Ei' jj sinh VI, 



(27) 



where I is the distance of the point in question from the e.m.f. Ex, 
and E x ' - Ex - SJ> 

It is only rarely that conditions in any transmission system 
would be definitely enough known to warrant the use of equations 
(26) and (27). Since all of the quantities entering these equations 
are complex, actual numerical computations from them are quite 
tedious, though not difficult. 



APPROXIMATIONS BASED ON NEGLECT OF LINE LOSSES 191 



116. Approximations Based on the Neglect of Line Losses. — 

In many cases it is not as important to know the actual values of 
the different harmonic voltages or currents along the line, as it 
is to be able to determine quickly the approximate maximum 
values which these quantities may attain. The presence of 
resistance and leakage in the line, leading to energy loss, serves 
always to prevent the attainment of infinite values of current and 
voltage, and, in general, serves to decrease the magnification of 
any particular harmonic, due to resonance conditions. Therefore, 
calculations made for a line with zero resistance and leakage, but 
with the same values of L and C as in an actual system, will yield 
values for the current and voltage distortion greater than could 
ever take place in the actual system, and thus serve as a guide in 
determining whether or not it would be necessary to make a rigid 
solution. 

Neglecting, then, the losses, 



r-#«j2*/ 



j8 = 2tt/VlC, and U 



-% 



so that, by cosh j(ft = cos (ft, and sinh j(ft = j sin {ft, equations (5) 
to (13) become, 



sin (k - V) 



'1 - 

Jz„=o 



sin $o 

cos g (jo - V) 

sinjSZo 



(28) 



E' 
V 



= E 



cos |8 (k - V) 



z =oo cos /3Zo 

-•jgi/jj fMft-*) 

="» V J, 



JZ { 



cos /3J 



(29) 



Eq = 



h = 



E 



cos /% + j y 7y Y sin /SZo 



E 



Z cos /3Zo + j y p sin pk 



(30) 



192 



CHANGE OF WAVE SHAPE ALONG LINES 
'sinpjlo- V) 



Jz =o " C cos/3/o 

j,l =/ co8<3(?o-n 
J z -o cos pk 



(31) 



E' 

r 



Z =<x> 



_ ., . /Z COS ft (*q - Q 

17 Vc sinftio 



= 7 



JZo«°° 

£ o =r 

7o = 



sin /3(^> — Q 
sinftio 



(32) 



lc 

Y cos Pk + j\j sin /3Z 



: — 7c — 

cos |3Zo + jZ y j sin /% 



(33) 



Z?0 

/o 



/o 



= E sec /SZo, 
Jz =» 

]= —jEK/t cosec/3Z . 
Z =0 » -tv 

= 7 sec jSZo- 



= —j? V ^ cosec Pt<>> 



jz =o 



(34) 



(35) 



For either short-circuited or open-circuited lines, computation 
by the foregoing approximate equations is very easy and rapid. 
It is not proper, of course, to use them when they reduce to either 
infinite or zero values, for then the line resistance and leakage 
become the predominating quantities which control the distribu- 
tion of line voltage and current throughout. In the simple cases 
of short-circuited and open-circuited lines, the equations may be 
used except for values of /3Z approximating multiples of 90 degrees. 
In this event, since for power lines a is approximately constant 
for frequencies above 60 cycles per second, the magnitudes of the 
phenomena at these critical frequencies may be easily determined 
from equations (5) to (13) by the simplified expressions, 



COMPARISON OF APPROXIMATE AND EXACT SOLUTIONS 193 

cosh ( al + j -z ) = j sinh al. 
COsh (al -\~jir) = — cosh al. 
cosh ial -\-j -tt-J = — jsinhaZ. 

sinh lal + j ~) = j cosh a£. 

sinh (al + j 7r) = — sinh aZ. }■ (36) 

sinh lal + j -~- j = — j cosh aZ. 

tanh (al+j-zj = coth aZ. 
tanh (al + j x) = tanh «Z. 
tanh f «Z + j —) = coth al. 

The above formulae are directly applicable to the discussion 
of quarter-wave transmission, where the frequency of the generator 
is adjusted to such a value that the line becomes a quarter wave- 
length. 

117. Numerical Comparison of Approximate and Exact Solu- 
tions. — Applying the approximations, equations (29), to the 
line whose voltage and current coefficients for the different fre- 
quencies are given in Table XXIV, the following numerical results 
are obtained, which, as approximations, may be compared with 
the accurate values given in Table XXIV. In computing the 
following, the value of 2 irf VLC is used for 0, and not the true 
values, dependent upon the line resistance and leakage as given 
in Table XIX. 

In the following table, the complex quantity representing the 
designated ratios is given in the form of a scalar and its angle. 

In all cases, except for a frequency of 420 cycles per second, for 
which the line is nearly a quarter wave length, the approximations 
are sufficiently close to the true results for all, or at least most, 
engineering purposes, and the labor required to secure the approxi- 
mate results is practically negligible in comparison with that 
required by the rigid expressions. 

For this line, by equations (13) and (36) when the frequency is 

slightly higher than 420 cycles per second, so that 0k is exactly x , 






194 



CHANGE OF WAVE SHAPE ALONG LINES 



the voltage at the open-circuited load end is 25.26 times the genera- 
tor voltage. 

TABLE XXV 

Approximate Values op Voltage and Current Coefficients from 

Equation (29) for a Distance ({') of 75 Miles from the 

Generator. 

Total line length = 100 Miles. True values of coefficients taken from 
Table XXIV for comparison. 





E' 




E' 




/' 




/' 




/ 


E 


, true 


-£-, approx. 


E 


10*. true 


E 1V ' 


approx. 


60 


1.020 


- 0°.44 


1.020 


0° 


0.140 


87°. 98 


0.141 


90° 


180 


1.211 


- 1 .52 


1.212 





0.504 


87 .90 


0.504 


90 


300 


1.871 


- 3 .61 


1.874 





1.316 


85 .98 


1.319 


90 


420 


6.895 


- 16 .60 


9.627 





6.944 


73 .04 


9.943 


90 


540 


3.192 


-171 .92 


3.228 


180 


4.262 


-82 .29 


4.310 


-90 


660 


1.319 


-176 .91 


1.322 


180 


2.242 


-87 .30 


2.247 


-90 


780 


0.878 


-178 .41 


0.879 


180 


1.855 


-88 .85 


1.857 


-90 


900 


0.717 


-179 .32 


0.718 


180 


1.867 


-89 .83 


1.868 


-90 



118. Resonance Frequencies. — In the general case, with the 
line closed through impedances at both ends, the frequencies at 
which resonance phenomena occur may be easily determined from 
equation (26). When resonance phenomena occur, the line cur- 
rents become very large, even for small impressed electromotive 
forces, and in the limiting case, when neither resistance nor leakage 
is present, and consequently no losses, the currents become infinite 
with finite voltages impressed. The only way in which these 
infinite values of Ii and I 2 can occur is for the denominator in 
equation (26) to become zero. That is, the values of / which 
cause the denominator of this equation to reduce to zero, when 
the line losses are assumed zero, are those at which the resonance 
phenomena occur. Thus we have for resonance, from (26), 

(Z, + Z 2 ) cos 0k + j (v | + V i ZiZ,) sin A = 0. (37) 

Zi and Z 2 are functions of the frequency, and though they may be 
made up of any combinations of inductances and capacities, no 
conductance or resistance should be present if the above equation 
is to apply rigorously. /3 a^so is a function of the frequency, 
/S = 2x/VXC. The solution of equation (37) for / must be 



RESONANCE FREQUENCIES 



195 



obtained by approximation — a graphical process similar to that 
used in determining the free frequencies of vibration for the 
oscillograph vibrator, Chapter V, Fig. 21, being convenient. 

To illustrate the use of this equation in determining the critical, 
or resonance, frequencies of a line connected as indicated in Fig. 65, 
consider again the same 100 mile power transmission circuit. 
Let one end be closed through an inductance of 0.050 henry, and 
the other through a condenser of 0.020 microfarad and an in- 
ductance of 0.10 henry in series. 

Then 

Zi=j27r/X 0.050, 

Z 2 =i(2./x0.10-^_). 



(38) 



For the line, per mile, 
L = 0.00204 



and 



C = 0.0146 X 10"*. 



Substituting these numerical values in (37) and expressing the 
result in such form as to utilize the tangent, we have, finally, 



tan0°.031269co = 



1 50 X 10 6 - 0.15 co 2 
co 7061.8 -0.000018376 co 2 ' 



(39) 



If we let y' represent the first and y" the second member of this 
equation, the intersections of the curves y' and y" plotted as 
functions of co will occur at those values of co which satisfy the 
above equation and which therefore are those corresponding to 
resonance conditions, or frequencies. These curves are shown in 
Fig. 66, from which intersections at the following values of co are 
noted: 



u, for resonance 


g>/2 x = frequency 


2,302 


366 


7,200 


1150 


12,400 


1970 


17,600 


2800 


19,700 


3130 


23,800 


3780 


29,300 


4660 


33,000 


5250 














196 



CHANGE OF WAVE SHAPE ALONG LINES 



Only the first two of these frequencies are sufficiently low to fall 
within the range of the higher harmonics produced by alternating 
current generators. It is interesting to note that the first reso- 
nance frequency, 366 cycles per second, is considerably lower than 

that for the open-circuited line alone, when /SZ becomes ~, or 90 
degrees, at a frequency of 458 cycles per second, 
as 







1 




1 














1 














3.0 






















1 

1 










1 


























1 










1 




2.5 




V 




1 














1 
1 


















\ 




1 

f 








II 






1 














2.0 




\ 














,i 






1 
1 






































1 










l 




1.5 


v' 






y '\ 






v'l 


' 






1 


i 






V' 










I 






t 

1 










i ! 






i 
i 










1 
J 




1.0 


» 






J 










1 1 
1 ! 


\y" 




/ 
















/ 
/ 






r 










/ ! 






1 








1 






0.5 


1 
1 






/ 






» 




/ ! 




V J 






/ 




1 

1 








1 
1 






/ 








, i 


f 












v' 


1 









1 

f 






/ 
f 




/ 




r 






/ 

/ 


<0 


sid 


■3/ 

1 




1 
1 













1 / 




8 


/ 
/ 


12 


0/ 




■ 


/ 


M 


V 


32 


i 


k 










/ 

/ 






/ 
/ 




/ 

/ 


\i 
ii 




/ 

/ 




' 


1 






-~ 



Fig. 66. — Graphical Determination of Resonance Frequencies from 
Equation (39). 

Even with this comparatively short line — 100 miles — and 
the small value of electrostatic capacity placed at one end of the 
circuit, this first resonance frequency of 366 cycles per second is 
only the sixth harmonic in a 60 cycle system. An increase in the 
capacity and self-inductance present would cause resonance at a 
still lower frequency, and since the fifth harmonic is quite prevalent 
in generator voltage wave shapes, resonance phenomena of this 
frequency may well appear in a one hundred mile transmission 
system. For line lengths between 200 and 300 miles, the first 
resonance frequency appears at about the third harmonic of 60 
cycles, or 180 cycles per second, so that resonance phenomena of 
this frequency may be produced if any source of e.m.f. of this 
frequency is present. 



RESONANCE FREQUENCIES 197 

From equation (37), used in determining the resonance fre- 
quencies in this particular case, it is apparent that it makes no 
difference whether the terminal impedances are inserted with Zi 
at the generator end and Z 2 at the load end, or vice versa. Res- 
onance phenomena appear at the same frequency in both cases, 
since Zi and Z 2 enter equation (37) only as sums or products. 

Even if the magnitudes of voltage and current produced by 
resonance condition for one of these higher harmonics should not 
be such as to endanger the power system, or even to be observed 
in their effect on normal operating conditions, they still may be 
of sufficient magnitude to create serious disturbance in communi- 
cating circuits which parallel the power system. For this reason, 
particularly, it becomes desirable, or necessary in some cases, to 
eliminate, as far as may be, the possibility of abnormal values of 
voltage or current of these higher frequencies in power systems. 
The large magnitudes of voltage and current of fundamental 
frequency are not so serious in their effect upon communication 
circuits as the smaller voltages and currents of the higher fre- 
quencies. 



CHAPTER X 
POWER RELATIONS IN LONG LINES 

119. Instantaneous and Average Power. — The instantaneous 
value of power at any point in a transmission line is equal to the 
product of the instantaneous values of voltage and current. In 
Fig. 31 and section 70 of Chapter VI is given an illustration of 
the instantaneous voltage, current, and power distribution along 
a line. The average value of power throughout a cycle is of more 
interest and importance. 

Explicit expressions (equations (3), (8), (18), (19), etc., Chapter 
VI) have been given for the voltage and current at any point in 
a line, and from these the equation for average power may be de- 
rived by taking the product of the scalar values of these quantities 
times the cosine of the angle of phase difference between them, 

P = ei cos <f>. 

By the relation that the cosine of the angle between two vectors 
is equal to the sum of the products of the direction-cosines of the 
two vectors, referred to two axes at right angles, a more convenient 
form of expression may be secured. Thus, referring to the e.m.f. 
and current represented analytically by 

E = d + jez and I = i x + fa, 
we have 

P = average power = ei cos <j> = e\i\ + e^h. (1) 

120. The Separate Components of E and / at any Point. — 

Obviously, then, the problem of setting up an explicit expression 
for the average power at any point in a line resolves itself into 
obtaining expressions for the separate components of E and I at 
any point and then forming the sum of the products of in-phase 
components. The details of such a development are quite simple, 
but nevertheless tedious. 

198 



THE POWER EQUATION 



199 



The quantities entering into the expressions for voltage and cur 
rent are 

V = a+j(3, 

U = Sj y = u/6 u = wi + juz, 



1 [Y ui .Wl . 

U = \-Z = ^- J u^ = yi + 



m- 



(2) 



(3) 



U 1 L W - U 

At the load end, let 

E = e f + to" and / = V + ji ". 

Then by equation (19), Chapter VI, the voltage and current at 
any point in the line distant I from the load end are given by 

E = (eo + to") (cosh al cos pi + j sinh al sin (il) 

+ (*»' + jV) ( w i + 7U») (sinh ai cos /3Z + j cosh aZ sin /3Z), 

1 = (i ' -|- fa") (cosh aZ cos /3Z + j sinh aZ sin fil) 

+ («b' + to") (Vi + J2/2) (sinh al cos 0Z ■+■ j cosh al sin #). 

These quantities are of the form 

E = ei+ to and / = i x + jt2, 
and by multiplying in equation (3), we have 

ei = e</ cosh aZ cos 01 — e " sinh al sin /3Z 
+ (Vwi — io'vv) sinh a? cos {il 
— (io'ih + to"wi) cosh al sin j8Z, 

e2 = e " cosh aZ cos /3Z + eo' sinh aZ sin (31 
+ (*o"wi + io'ik) sinh aZ cos /3Z 
+ (io'ui — io'vv) cosh al sin /3Z. 



(4) 



t'i = tV cosh aZ cos 01 — to" sinh aZ sin /3Z 
+ (eo'2/1 — eo"y2) sinh aZ cos jSZ 
— (eo'2/2 + eo"!/!) cosh al sin 0Z, 

t2 = t'o" cosh al cos /3Z + V sinh aZ sin /3Z 
+ (eo"t/i + eo't/2) sinh al cos /3Z 
+ (eo'yi — e "y2) cosh aZ sin /3Z. 



(5) 



121. The Power Equation. — Substituting the above expres- 
sions for the separate components in the equation 

P = 01*1 + Ste, ( 6 ) 



200 POWER RELATIONS IN LONG LINES 

multiplying out, and finally combining terms, there results the 
following form: 

P = ( eo V + eo'%") (cosh 2 al cos 2 01 + sinh 2 al sin 2 01) 

+ [(eo' 2 + eo" 2 ) yi + (to' 2 + V 2 ) u,] cosh al sinh aZ 
- [(eo' 2 + eo" 2 ) v, + (V 2 + to" 2 ) u*] cos 01 sin /3Z 
+ [(eoV + eo"to") (u lVl + u*y 2 ) + (eoV - eo'V) 
(Mit/j — M2?/i)] (cosh 2 aZ sin 2 01 + sinh 2 a£ cos 2 01). (7) 

Considerable care is required in reducing the product as indicated 
by equation (6) to the form given in (7) on account of the large 
number of terms involved. 

The power at the load end, or receiver, is 

Po = e V + e "to" 

and the scalar values of voltage and current at the load end are 

e^ = eo'2 + ^ and tf = V 2 + to" 8 . 

Introducing these values into equation (7), 

P = Po (cosh 2 al cos 2 01 + sinh 2 al sin 2 01) 
+ (eo^i + t'o^i) cosh al sinh aZ 
— (eo 2 ^ + to 2 ^) cos 01 sin /5Z 
+ [-Po (miVi + M22/2) + (eo'to" - eo'V) 

(M12/2 — M22/1) ] (cosh 2 al sin 2 /3Z -f sinh 2 aZ cos 2 /3Z) . (8) 

122. Reduction in Form of the Power Equation. — Still fur- 
ther reduction is possible by a consideration of the necessary 
relations existing between ui and M2, and 2/1 and y 2 . 



u 


= Mi + JMa 


= y y = m/0 u , m 


2 = Mi 2 + M2 2 . 


1 


= yi + jt/2 • 


Mi 


M2 


u 


Mi 2 + M2 2 J M1 2 


+ Ma 2 




Vi = 


Mi , 

5 and y 2 = - 


M2 
M 2 ' 


Mit/i 


+ Mat/2 = 

Mi 


2 — M2 2 _ - M2 2 

2 + M2 2 M 2 


= 1 + 2 M22/2 


uiy 2 


- M2V1 = - 


0M1M2 . 

2— 5- = +2mi2/ 2 . 





Thus 



and (9) 



REDUCTION IN FORM OF THE POWER EQUATION 201 

Introducing these expressions into equation (8), 

P = P (cosh 2 al cos 2 pi + sinh 2 al sin 2 pV) 
+ (eo 2 Vi + io 2 u{) cosh al sinh aZ 
— (eo 2 2/2 + *o 2 t*2) cos j8Z sin pi 
\Po -f 2 [P0W22/2 + (eoV - e</V) uiy*]\ 

(cosh 2 aZ sin 2 0Z + sinh 2 al cos 2 01) . (10) 

Combining the first term, and the portion of the fourth term 
in the right hand member which contains Po alone, there results 

P [cosh 2 al (cos 2 pi + sin 2 pi) + sinh 2 al (cos 2 pi + sin 2 pi)], 

which reduces to 

Po (cosh 2 al + sinh 2 al) or P (1 + 2 sinh 2 al). 

The expression 

*%" - e "io' = P i (11) 

represents the wattless volt-amperes at the load end (see Stein- 
metz' "Alternating Current Phenomena," page 218), and is 
commonly designated by the symbol, PK Numerically, also, 

Po ? '= eo^sin 0o, 

where <£o is the power-factor angle of the load, counted, in the 
vector diagram, from E to I in a counter-clockwise direction, and 
is thus positive in value for a leading (condensive) load. The real 
power is 

Po = eoio cos <f>, 
so that 

volt-amperes = e «o = Vp 2 + P > 2 . 

In further writing, products of the form 

UV COS (0„ — 0„) = U1V1 + W2V2 

may be written in the abreviated notation, U • V; products of 
the form 

uv sin (0„ — 8 U ) = U1V2 — U2V1 

may be written with the notation, U X V. 

U . V ■ UV COS (&x — 6v) = U\Vi -f W2«>2, 

£7 X 7 ■ uv sin (0„ — 0„) = U1V2 — U2V1, 



202 POWER RELATIONS IN LONG LINES 

where 

U = Ui + jut, V = Vx + JUj, 
UV ss (u i+jifr) ( vi +jihh_ 
uv = Vui 2 + U3 2 Vy x 2 + v, 2 . 



Therefore 



(uvy = (U-V)*+(Ux V)\ 



For example, 

P = E'l, P' = E X /, and volt-amperes = ei. 

123. Final Form of Power Equation. — For further reduction 

cosh al sinh al = \ sinh 2 aZ, 
cos/3Zsin/3Z = 5sin2j8Z, 
and 

cosh 2 al sin 2 /SZ + sinh 2 aZ cos 2 /3Z = sinh 2 aZ + sin 2 /3Z, 

so that the final form may be taken to be 

P = P (1 -f- 2 sinh 2 al) + (eo^i + *o 2 Wi) sinh al cosh aZ 
— (eo 2 ^ + *o 2 M2) sin /SZ cos # 
+ 2 (P0U22/2 + Po'ttijfc) (sinh 2 «Z + sin 2 /SZ). (12) 

124. Numerical Example. — As an illustration of this final 
equation, let us compute the power at- the generator for a 100 mile 
transmission line with the constants 

r = 0.275, g = 0.15xl0-«, L= 0.00204, and C= 0.0146 Xl(H 

per mile, when the e.m.f. and current at the load end are respec- 
tively 100,000 volts and 200 amperes, and the load power-factor 
angle is <£o = —25.0 degrees. This signifies a "lagging load." 
Let the frequency be 25 cycles per second, and in order to use the 
equation in its general form, let neither Eo nor Jo be selected as 
reference vector, but assume 

Eo - eo' + jeo" = 100,00 0/65.00 deg. = 42,262 + j 90,631 volts, 
r- U' + jV = 200.0 /40.00 deg. = 153.208 + j 128.558 amp. 

Then 

Co = 100,000 volts, 
to = 200.00 amperes, 
^o = - 25.000 degrees, 



NUMERICAL EXAMPLE 203 

and 

Po = eo'io' + e"W = eoio cos <£o = 18,126,300 watts, 
iV = e<>%" — eo"io' = e<#o sin <f> = — 8,452,400 volt-amps., 



tf-v/f- 



- Mi + JM2 = 406.61 — .; 135.63 ohms, 
jj = y ^ = ft +jy 2 = 0.0022131 + j 0.00073821 mhos. 

From Table XIX, 

al = 0.037205, 

&l = 5.2264 degrees. 

Substituting these numerical values in equation (12), the four 
separate terms which go to make up the total power at the genera- 
tor, P, are, in consecutive order: 

First term + 18, 176,700 watts 

Second term + 1,429,810 " 

Third term - 177,520 " 

Fourth term - 84,280 " 



P = 19,344,710 watts. 

Computing the generator e.m.f. and current by equation (19), 
Chapter VI, we have 

E = 41,892 +.7*98,894 volts, 
J = 132.10 + j 139.65 amperes, 
and the power 

P = eiii + e 2 * 2 *" e * cos = 19,344,500 watts, 

which verifies completely the value obtained from equation (12) 
If the line loss be roughly computed by 

P' = ioh-l + e<?gl, 

we obtain P' = 1,250,000 watts, while the true value of the line 

loss is 

P' m P - P = 1,218,410 watts, 

the last three figures of which are unreliable, since the computa- 
tions were made with five-place logarithm tables. 



204 POWER RELATIONS IN LONG LINES 

125. Line Loss. — For very accurate work, it may be desirable 
to use the formula 

P' = P-P = P 2sinh 2 aZ 

+ (eo 2 2/i + io 2 Ui) sinh al cosh al 

— (eo 2 2/2 + to 2 M2) sin 01 cos 01 

+ 2 (P U2i/2 + Po y Uiy f ) (sinh 2 al + sin 2 0Z), (13) 

to calculate the line loss directly, for this obviates the numerical 
inaccuracy involved in taking the difference between two large 
and nearly equal quantities, in the case of a line whose losses are 
small in comparison with the power transmitted. With long 
telephone circuits, on the other hand, the loss in the line may be 
many times the power received at the load end, so that values of 
power loss for a given load may be conveniently obtained either 
by numerically subtracting, P — P , from equation (12), or by the 
use of (13) where the subtraction has been made in the analytical 
expression. 

126. Unloaded Line. — For an unloaded line, P = 0, the 
expression for P is also the expression for line loss, and it is 

P = {&?V\ + fco 2 Wi) sinh al cosh al 

JPo=0 

— (eo 2 2/2 + io 2 ih) sin 01 cos 01 

+ 2 Po'Wii/2 (sinh 2 al + sin 2 01) . (14) 

The equation as given may be applied to a line supplying a load of 
zero power-factor, so that P = 0. Po } is then numerically equal 
to eo^o — positive for a leading load and negative for a lagging 
load. If both P and P y are zero, then the line must be either 
short-circuited or open-circuited (e = or in = 0), and for these 
conditions, equation (12) becomes 



P = to 2 (wi sinh al cosh al — ih sin 01 cos 01) , 

Jeo=0 

P\ — ^ (yi sinh al cosh al — t/ 2 sin 01 cos 01) . 

J«o=0 



(15) 



127. Power in Distortionless Lines. — In Chapter VII, the 
quantity V = a + j0 was discussed, and by equations (13) and 
(14) it is seen that for all frequencies, the attenuation constant, 
a, and the velocity of propagation, 8, are constant, and the wave- 



POWER IN DISTORTIONLESS LINES 205 

length constant, /3, is directly proportional to the frequency when 
the relation 

gh = rC (for distortionless line) (16) 

is fulfilled. A line whose properties are thus related is said to be 
a "distortionless line." The reason for this name is quite obvious 
— the constancy of a and S for all frequencies. If an alternating 
wave is made up of a number of harmonic frequencies, all of these 
different frequencies in either an e.m.f. or a current wave will be 
propagated along such a line with the same attenuation and the 
same velocity. Hence the wave form of the propagated wave 
does not change, all of the component harmonics retaining the 
same relative phase position throughout the line, and all decreasing 
in the same proportion. This, of course, is true only for the com- 
ponent (main and reflected) waves, and not for the resultant wave, 
except in the case of an infinitely long line, where the main wave 
and resultant wave are identical. This matter of wave distortion 
is of the utmost importance in telephonic work, and will be dis- 
cussed in connection therewith. 

In such a distortionless line, then, by equation (16), 

g r 
where k is the proportionality constant. Then 

Z = r+jwL = r + jwkr, J . * 

Y = g+ju } C = g+ja>kg,\ KU) 

so that 

"-v/f-vMBI-v^ 

This equation, interpreted, shows that for a distortionless circuit, 
the infinite line impedance, U, contains no imaginary component, 
and has a numerical value equal at all frequencies to the infinite 

line resistance for direct current, y - • 



U = U1+JM2 = V"' 

JgL-rC ' g 

y] = yi +jy 2 =\fl, 

JgL=rC Y T 

uJ = 0, y 2 \ =0. 

JgL=rC JgL=rC 



(18) 



206 POWER RELATIONS IN LONG LINES 

Introducing this condition, that both ut and y 2 are zero, into the 
general equation for power, (12), there results, for the distortionless 
line, 



J l - 

JgL-rC 



Po (1 + 2 sinh 2 aZ) f . 

+ ( e ° 2 \~ r + *° 2 V ") sinh al cosh al > ( 19 > 



a very simple and interesting form. Since 

cosh 2 al = 1 + 2 sinh 2 al 

it is seen that equation (19), for a load power-factor of unity, is 
identical with equation (26) of Chapter II for a direct current line. 
In this case, as has been shown to be true for any frequency, 



-1 " 



so that, finally, 

p] - Po cosh 2 VgTl + ± (e 2 V Q - + io 2 y -) sinh 2 V^Z. (20) 

This resulting value of power at the generator end for a given 
power, Po, at the load end, is entirely independent of the frequency, 
in so far as the fundamental line constants r, L, g, and C are in- 
dependent of the frequency. 

Consider the numerical case used as illustration of equation 
(12) in the last paragraph. For this line, gL = 0.0003060 X 10"« 
and rC = 0.0040150 X 10 -6 . Suppose g, r, and C to remain 
constant, but by some means the value of L is increased until the 
relation, gL = rC, is fulfilled. The line then becomes distortion- 
less, and equation (20) applies. The self-inductance, L, must be 
increased from 0.002040 to 0.0267667 henry per mile to bring 
about this condition. Then, for the same load for which the 
preceding numerical calculation was made, we have for the power 
at the generator, by equation (20), 

P = 19,391,400 watts 
and 

P' = 1,265,200 watts = line loss. 

The line loss with the original value of self-inductance per mile 
was 1,218,400 watts, indicating that, in this case, the creating of 



LOAD IMPEDANCE FOR A MINIMUM LINE LOSS 207 

a distortionless circuit by an increase of the self-inductance does 
not bring about a decrease in the line power loss. From this 
particular numerical example it is not possible to generalize, 
however, to the extent of saying that the creation of a distortion- 
less line by an increase in the self-inductance will always cause an 
increase in the power loss for a given load. It is possible to 
differentiate equation (13) with respect to either L or C (remember- 
ing that the auxiliary constants, Ui, Vq, y\, yi, a, and /3, are func- 
tions of r, g, L, and C), and thus to determine for what values of 
the independent variable the line loss would be a minimum, but 
the resulting expression after differentiating would be so com- 
plicated as to prevent any practical results from being obtained. 
For a given load, and either L or C variable, it would be best to 
plot a curve showing the power loss as a function of the variable, 
and from the curve to estimate the value for a minimum loss. 

128. Load Impedance for a Minimum Line Loss with a Given 
Po. — In transmitting power over a leaky line by means of a 
direct current, a maximum efficiency of transmission occurs when 
the load-end resistance (equivalent resistance of the load) is equal 

to the infinite-line resistance y -, according to equation (34), 

Chapter II. With this value of load resistance, the efficiency of 
transmission is constant and independent of the magnitude of 
voltage and current. 

A similar condition obtains in the case of a line carrying alter- 
nating currents. As an introductory case, suppose an amount 
of power, P , is to be delivered at the load end of a line, at a power- 
factor, cos <f>o. This constant amount of power may be supplied 
by current at any voltage, the necessary relation being 



eoio cos 0o = -Po, or e io = 



Po 



COS 0o 



What must the values of e and i be in order that the line loss be 
a minimum? 

Let 2 = -r = scalar value of load-end impedance. 
to 

Then 

if = — ^— and e 2 = ^» (21) 

Zocos<£o COS 0o 



208 POWER RELATIONS IN LONG LINES 

Since P and fa are assumed constant, P y = P tan </>o is also a 
constant. 

Thus, if W represent the variable portion of the line loss as eo 
and io are varied, with constant Po and <&, we have, from equation 
(13), 

W **\ (eo 2 yi + io 2 Mi) sinh 2 al - £ (eo 2 y 2 + t'o 2 ^) sin 2 /3Z, 
and by equation (21), 

W = ^(-y^-+— ^-)sinh2aZ 
2 \cos0o Zo cos 0o/ 

_Po/_^. _J^.\ in ^ ( 

2 \COS0o ZoC08(f>o/ v ' 

In order that this variable portion of the line loss, and thus the 
total line loss, be a minimum, differentiate equation (22) with 
respect to Zq and equate the derivative to zero. 

dW Po / Mi\ . , „ , 

^ = T sec<^-- 2 )sinh2aZ 

- ^ sec <t> (y 2 - §1 sin 2 0Z = 0. (23) 

Solving (23) for z , 

2 _ U\ sinh 2 aZ — tt2 sin 2 01 _ U\ sinh 2 aZ — iiy sin 2 ffl 2 ,_ .. 
2/i sinh 2al — y 2 sin 2 $ Mi sinh 2 a£ + W2 sin 2 /3i 

The value of Zq given by equation (24) is independent of the 
amount of power delivered, as well as of the load power-factor. 
That is, for a given amount of power delivered, to secure a mini- 
mum line loss under any condition, the load impedance should 
have the above numerical value. For varying amounts of deliv- 
ered power and varying power-factors, such a condition demands, 
therefore, a variable and varying generator voltage — a condition 
which is usually not feasible, in power transmission at least. 
Notice that, in regard to the above statement of minimum line 
loss under any condition, an absolute minimum of line loss is not 
implied. For a given amount of power at a given power-factor 
the line loss is a minimum when «o is determined by the above 
equation, but obviously, by equation (13), for a given power Po 
and load impedance Zo, the line loss may be changed by varying 
the power-factor angle, <ft>, of the load. 



NUMERICAL ILLUSTRATION 209 

129. Numerical Illustration. — As illustration of equation (24) 
consider the 400 mile, 60 cycle line, whose constants are given 
in Tables XII and XIV and some operating characteristics of 
which are given in Tables XV and XVI. A summary of the con- 
stants, and the calculations leading to a value of Zq by equation 
(24) is given below in Table XXVI. 



TABLE XXVI 
Determination of z for Minimum Line Loss, by Equation (24) 



r 


. 275 ohm per mile 


I 


0. 15 X 10 -6 mho per mile 


0.00204 henry per mile 


C 


. 0146 microfarad per mile 


I 


400 miles 


f 


60 cycles per second 


2 at 


0.312615 


2 01 


95.5384 degrees 


U = Ui +ju 2 


385.143/-9 03' 26". 9 = 380.34 -j 60.630 


ij = yi +jyt 


0.00259644/9° 03' 26". 9 - 0.0025641 +j0. 00040875 


sinh 2 al 


0.31773 


sin 2 01 


0.99534 


Mi sinh 2 al 


120.84 


u 2 sin 2 01 


-60.35 


j/i sinh 2 al 


0.00081468 


y 2 sin 2 01 


0.00040684 


Numerator 


181.19 


Denominator 


0.00040784 


zo 2 


444260 


Zo 


666.53 ohms 



It thus appears that, for this line, a minimum line loss will be 
secured when, for any load, the generator voltage is so adjusted 
that the ratio, Zo, between the numerical values of load voltage 
and load current is 666.53. For example, to supply a load of, say, 
4000 kilowatts at a power-factor of cos 25 degrees = 0.9063, the 
load voltage should, by (21), be 

eo = V *<> t = V 666.53 ' ' = 54,238 volts, 

▼ cos <£o v 0.9063 

and the corresponding load current is 

zo = — = 81.373 amperes. 

Zq 



210 POWER RELATIONS IN LONG LINES 

Assuming this to be a lagging load, we have, with fa = —25.000 
degrees and P = 4,000,000 watts, 

Po' = Po tan <fo = - 1,865,200 volt-amperes. 

By equation (13), 

P' = line loss = 951,030 watts. 
P = power at generator = 4,951,030 watts. 

p 

Efficiency of transmission = y = -£ = 0.8079, or 80.79 per cent. 

Note that this value of efficiency corresponds approximately 
with that for the 80 ampere entry in the first part of Table XV. 

130. Line Efficiency. — As before pointed out, if the load 
impedance remain fixed, the current and voltage throughout the 
line change directly as the generator voltage. All powers and 
volt-amperes then vary as the square of the generator voltage, so 
that the efficiency remains constant. Since for any value of power, 
Po, a minimum line loss, and therefore a maximum efficiency of 
transmission, is obtained with the above value of Zo, it follows that 
each particular power-factor of load has a definite and character- 
istic maximum possible efficiency, viz., that corresponding to the 
value of Zo by equation (24). A curve may thus be plotted show- 
ing the maximum possible efficiencies as a function of the load 
power-factor, and such a curve is independent of the line voltage 

employed. 

p 

Let r] = efficiency of transmission = -^ and Z = Z p/— <fo . 

The angle of Z is taken negatively because an impedance with 
a positive angle, r + jx = z/^ , corresponds to a negative power- 
factor angle (lagging) load, and vice versa. In this discussion, 
<fo is the power-factor angle — the angle of 7 with respect to E . 
Then, by equations (12) and (21), 

— =1 + 2 sinh 2 al + 2 (uzy2 + Uiy 2 tan <fo) (sinh 2 al + sin 2 fil) 

+ 2^b [(■* + 1) sinh 2 al ~ (*» + 1) sin 2 "J • (25) 

If </>o be kept constant in the above equation for efficiency, and 
the first derivative with respect to Zq be equated to zero in order 
to determine the value of Zo for either a maximum or a minimum, 
the result given by equation (24) will be obtained. Substituting 



LOAD POWER-FACTOR FOR MAXIMUM EFFICIENCY 211 

(24) for Zq in equation (25), the efficiency expression for this special 
load impedance becomes: 

—1 =1 + 2 sinh 2 al + 2 Uiy 2 (sinh 2 al + sin 2 01) 

V Jzo by (24) 

+ 2 U\\)i (sinh 2 al + sin 2 01) tan 0o 

— - Vuij/i sinh 2 2<xl + Utfji sin 2 2 01 (26) 



COS 0o 

131. Load Power-Factor for Maximum Efficiency. — Differ- 
entiating with respect to 0o, 

'ft) 

j = 2 Wi?/2 (sinh 2 aZ + sin 2 01) sec 2 0o 
o<Pq 

+ sec 0o tan 0o Vui2/i sinh 2 2al + wyi sin 2 2 01. 

Equating the first derivative to zero, in order to determine 0o for 
a maximum or a minimum, and solving, 

2M 1 y 2 (sinh 2 o!? + sin 2 j3Z) 

sin 0o = 7 ==> (27) 

vuiyi sinh 2 2 a J + W22/2 sin 2 2 01 

for a condition of maximum efficiency, when taken in connection 
with the value of Zq by equation (24). To be rigorous, it is neces- 
sary to determine that the value of 0o by (27) corresponds to a 

minimum of — and thus to a maximum efficiency, rj. For 

such to be the case, the second derivative of — with respect to 0o 

v 
must be positive for the value of 0o given by equation (27). 

Differentiating again, 

*(t) 

, = 2 Uiy 2 (s'inh 2 al + sin 2 01) 2 sec 2 O tan O 

U0o 



+ Vinyi sinh 2 2 al + 1*22/2 sin 2 2 01 (sec 3 O + tan 2 O sec 0o) 
= sec 3 0o { 4 M1I/2 (sinh 2 al + sin 2 /3Z) sin O 
+ (1 + sin 2 0o) Vmyi sinh 2 2al + my* sin 2 2/8Z}. 
Substituting for sin 0o the value given by (27), 



(P 



(7) 



«6 by (24) 
<fr> by (27) 



d0o 2 

[Mii/i sinh 2 2 al + v^yt sin 2 2 01 - 4 1*1 V (sinh 2 al + sin 2 #) 2 } 

sec 3 0o 

Vuiyi sinh 2 2 al + W22/2 sin 2 2 01 



(28) 



212 POWER RELATIONS IN LONG LINES 

It remains to be demonstrated that the value of the expression 
in equation (28) is always positive. From purely physical con- 
siderations, we know that if there are no line losses at all, the 
efficiency of transmission will always be unity, and therefore there 
will be no maximum or minimum and the second derivative as well 
as the first derivative have zero values. When there are line losses, 
the efficiency must reach a maximum value somewhere between 
its two zero values corresponding to leading and lagging loads of 
zero power-factor, but of the proper impedance Zq to make equation 
(28) applicable. Sin fa by equation (27) must then be real, from 
which we know that the quantity under the radical must be 
positive. The value of the radical itself is essentially positive, 
since the scalar value of Zq by (24) must be positive. Sec 3 fo is 
always positive, so the value of the second factor in (28) is always 
positive. The quantity in the brackets remains in question. 

To show that the quantity in brackets is always positive appears 
rather complicated, but some special cases covering the widest 
possible range of conditions possible in transmission lines may be 
investigated with ease. 

For a distortionless line, gL = rC, the value obviously is positive, 
since by (17) u^ and 1/2 both vanish, and Mi and y\ are positive. 

Again, suppose that g = and L = 0, a condition represent- 
ing maximum distortion. Then 

Z = r+j0 and Y = +j2*fC = + jb. 



vy V b c — =- V 26 v 


1 


» - * - v Tr 


1 


V = a + j/3 = Vjrb = Vrb/45° = ^ 


/t+ivf 


«-,-vf 




Then 





uiyi = \, ikyi = -\, and UiV = \. 
Substituting these values in equation (28) and denoting by B the 



NUMERICAL ILLUSTRATION 213 

value of the expression in brackets, we have, when x represents the 
common value of al and /3Z, 

B = % (sinh 2 2 x - sin 2 2 x) - (sinh 2 x + sin 2 x) 2 . 
Expanding in a series, 

The coefficients of the series all being positive, the value of B is 
always positive, for this special case, and therefore the value of 

the second derivative of — is positive. 

V 

If we take the other extreme, that is, r = and C = 0, the same 
result will be obtained. 

Since for the above three conditions as widely separated as 
possible the value of <£o given by equation (27) corresponds to a 
maximum value of transmission efficiency as desired, it is reason- 
able to assume that the equation is correct under all conditions. 
Several numerical examples with intermediate conditions have 
shown this to be true. 

132. Numerical Illustration. Variation of Efficiency with 
Power-Factor. — Applying equation (27) for O to the numerical 
case under discussion, see Table XXVI, the computation appears 
as below, in Table XXVII. 

TABLE XXVII 

Determination of <t>o foe Maximum Efficiency of Transmission 
Values from the logarithmic computation for Table XXVI 

log uiyi sinh 2 2 at 8 . 99322-10 

* log u 2 y 2 sin 2 2 /3Z -8 . 39008-10 
log sum of above 8.86864-10 
log denominator 9.43432-10 
log numerator 9.25072-10 

* log sin 0o -9 . 81640-10 

*o -40.938 degrees 

cos<fc> 0.75542 

* The minus sign before the logarithm does not signify a negative logarithm, but that the num- 
ber represented thereby is negative. 

The angle of Zo is of opposite algebraic sign from that of <fo, so 
that 

Zo = Zo cos <£o — jzo sin #o. 



214 



POWER RELATIONS IN LONG LINES 



From Table XXVI, «o = 666.53 ohms, which gives Z = 503.51 +j 
436.74 ohms, for a condition of maximum efficiency of trans- 
mission. 

It thus appears that, for this line, a maximum efficiency of 
transmission is obtained when the load power-factor angle is 
—40.938 degrees; that is, for a lagging load having a power-factor 
of 75.542 per cent. The numerical value of the load impedance 
being fixed by equation (24) and Table XXVI, the separate com- 
ponents thereof are determined as indicated in Table XXVII. 

n = 503.51 and x = 436.74 ohms. 

With this value of load end impedance, the efficiency always 
remains constant at its maximum value, regardless of the gener- 
ator voltage employed, and the power received varies, of course, as 
the square of the generator e.m.f. The efficiency for the above 
value of Zq is, by equation (25) or (26), 81.550 per cent. Numeri- 
cally, equation (26) is 

— = 1.020869 + 0.17812 tan <fo + 0.27184 sec 0o, 



or 

1 

9 1.020869 + 0.17812 tan O + 0.27184 sec *,' 

from which the following values were obtained: 



(29) 



TABLE XXVIII 

Transmission Efficiency of 400 Mile, 60 Cycle Transmission Line 

z = 666.53 ohms. Values from equations (26) and (29) 
<I>q = load power-factor angle. »? = line efficiency 



<t>0 


V 




deg. 






-90 


0.00000 




-80 


0.63444 




-60 
-40.938 


0.79615 
0.81550 


Lagging 


-40 


0.81547 




-20 


0.80301 







0.77357 




20 


0.72728 




40 
60 


0.65565 
0.53388 


Leading 


80 


0.27804 




90 


0.00000 





NUMERICAL ILLUSTRATION 



215 



Fig. 67 shows the curves plotted from the above data. It is to 
be seen that throughout the possible ranges of power-factor, a 
much better efficiency of transmission is obtained with a lagging 
load than with a leading load. For the quite common power- 
factor angle of 40 degrees (P.F. = 0.766), the efficiency of trans- 
mission has a value with a lagging load of 81.55 per cent, while at 
this same power-factor and a leading load, the efficiency is only 























































































































































V 










































1.0 










































0.0 










































0.8 










































oT 










































0.0 










































0.5 










































0.1 










































0.3 










































0.2 






























Lbj 


Sin 


I L 


JlKl. 






0.1 




I 


.eai 


ing 


Lou 


d. 














I 


'owe 


L*4* 


igle 













D 


«W 


es. 












-90-80 


-60 

1 


-40 

1 


"t 


-0 

1 


f 


40 

1 


T 


80 & 



Fig. 67. — Transmission Efficiency with 2o Constant at its Maximum- 
efficiency Value. Curve Plotted from Equation (29). 

65.56 per cent — a difference of 16 per cent. This being the case, 
it is obvious that from the standpoint of line losses, or efficiency, 
a lagging load is much to be preferred to a leading load, although 
in practice, the latter is viewed with favor on account of the superior 
voltage regulation obtained thereby. This is a direct result of 
the use of inherently constant voltage apparatus, and the conse- 
quent desirability of maintaining as nearly as may be a condition 
of constant voltage transmission. Theoretically, the ideal con- 
dition of transmission would be that for which the load power- 
factor would remain constant, while the voltage, as well as the 
current throughout the line, would vary as the square root of the 
power transmitted. From such a condition, a maximum economy 
in the use of power would result. 



216 POWER RELATIONS IN LONG LINES 

For this particular line, then, the ideal conditions (as far as 
efficiency is concerned) may be summarized in a number of equa- 
tions, as follows: 

From equation (24) and Table XXVI, 

zo ■ 666.53 ohms. 
From equation (27) and Table XXVII, 

0o - -40.938 degrees, and Z = 503.51 + j 436.74 ohms. 
From equation (26) and Table XXVIII, 

r] = efficiency of transmission = 0.81550. 

From equation (21), 

to = 0.044565 VPp and eo - 29.704 VW . 
At the generator, or sending end, 

p = f?= 1.2263 P . 

v 

From equation (19), Chapter VI, and Table XIV, 
i = 0.049348 VF and e = 32.894 Vp . 

i = \/J-i = i.i073to and e = J—eo = 1.1073 eo. 

From equation (19), Chapter VI, 
4> = power-factor angle at the generator = +40.938 degrees. 

m 

z = - = apparent generator-end impedance = 666.53 ohms. 

I 

133. Peculiarities Incident to Maximum-efficiency Trans- 
mission. — An inspection of these numerical values reveals the 
very interesting fact that, for a condition of maximum efficiency, 
the power-factor angle at the generator end is equal in magnitude, 
but of opposite algebraic sign from the power-factor angle at the 
load. The apparent impedance at the generator, z, is numerically 
equal to the load impedance, Zo. From load to generator, both 
current and voltage increase by the same percentage, and from the 
above, the ratio of similar quantities is seen to be 



'- - i - t/i- 

eo to '.% 



VARIATION OF MAXIMUM-EFFICIENCY CONDITIONS 217 

The above statements have been taken from the numerical values 
for this particular line only, and though .the agreement of the 
numerical values through five significant figures is sufficient to 
establish their correctness beyond any reasonable doubt, general 
proof will be given later. (See section 138.) 

134. Variation of Maximum-efficiency Conditions with Line 
Length. — The following table shows the numerical values 
pertaining to a condition of maximum efficiency of transmission 
over lines possessing the same fundamental constants per unit 
length as used in the above numerical illustration and of the 
various lengths tabulated. The constants are: / = 60 cycles, 
r = 0.275, g = 0.15 X 10^, L = 0.00204, and C = 0.0146 X 
10 -6 . For convenience in determining the efficiency for load power- 
factors other than those corresponding to maximum efficiency, the 
constants entering into equation (26) are tabulated. They apply 
to equation (26) when it is written in the form, 



— = A + B tan <6 + C sec <*o. 
v 



TABLE XXIX 

Conditions Obtaining when Lines op the Previously Given Constants 

and of the tabulated lengths are operated at the 

Maximum Efpichcncy Possible 









Constants in Eq. 


26 






zo. ■ 
ohms 


0Ot 

degrees 










Miles 








V 








A 


B 


C 







1354.0 





1.0000 








1.00000 


100 


1243.9 


-18.411 


1.0008574 


0.013788 


0.04366 


0.95944 


200 


1023.9 


-31 . 175 


1.003812 


.052876 


. 10214 


.91641 


300 


821.3 


-38.092 


1.009946 


.11082 


.17964 


.86857 


400 


666.5 


-40.938 


1.020869 


. 17812 


.27184 


.81550 


600 


552.8 


-41.054 


1.03845 


.24386 


.37129 


.75848 


600 


469.7 


-39.184 


1.06454 


.29753 


.47091 


.69952 


800 


367.8 


-31.066 


1 . 14780 


.33942 


.65777 


.58438 


1000 


328.4 


-19.842 


1.27570 


.28583 


.84210 


.48361 


1200 


334.1 


-10.052 


1.44355 


.18453 


1.05731 


.40247 


1400 


365.6 


- 4.110 


1.64193 


.11800 


1.30761 


.33962 


1600 


399.9 


- 5.431 


1.86532 


. 14971 


1.58168 


.29071 


1800 


416.5 


- 8.616 


2.1188 


.28303 


1.8892 


.25083 


2000 


409.8 


-11.790 


2.4183 


.45958 


2.2493 


.21645 


Inf. 


385.1 


- 9.057 


Inf. 


Inf. 


Inf. 






218 



POWER RELATIONS IN LONG LINES 



% 


20 

liog 




























































































1200 










































*0 














































1.0 


1000 










































-40 


















x' 




























-3B 


M 


800 
























































3 






























-28 


O.r, 


600 
























































































-20 


O.J 


WO 












r> 










































































-16 










-12 
-8 


o.e 


■200 










































































































Liiu 


U- 


a*q 


i, M 


lies. 


















—4 







a 





* 


HJ 


600 


800 

1 


1000 


1200 


1400 


1600 


1800 


20 Xf 



Fig. 68. — Values Obtaining Under Maximum-efficiency Conditions. 



A 

and 












































c 

2.6 






























i. 












B 

0.52 


2.4 










































48 


2.2 










































0.44 


2.0 










































0.40 


1.8 








































/ 


0.88 


1.6 


















«J 
























0*J2 


1.4 


























hy> 






'fc 










n-?fi 


1.2 










































n?4 


1.0 












































as 










































0.16 


0.6 










































cm 


a4 










































0.08 


02 










































0.04 



















Li 


le I 


eog 


thi 


i M 


lcs 





















J 


1 


X) 


i 


1 


I 


* 


800 


1000 1 1200 

1 1 1 


14 


JO 


vm 


1800 


a 


. 



Fig. 69. — The Constants in the Equation for Efficiency, 
1 

n 



A + B tan <to + C sec <fo. 



ZERO LENGTH LINES 219 

Curves plotted from Table XXIX are shown in Figs. 68 and 
69. In Fig. 68, for continuously increasing line length, the 
numerical value of z as shown for a condition of maximum effi- 
ciency decreases from a finite maximum at zero length to a finite 
value at infinite length. On the other hand, O , the load power- 
factor angle for maximum efficiency, increases rapidly in numerical 
value from zero for a fine of zero length to a maximum of about 
—41 degrees and then decreases, with a two-sided approach to 
a limiting value for infinite fine length. The maximum possible 
efficiency decreases continuously, approaching zero ultimately. 

135. Zero Length Lines. — Special formulae are required for 
the determination of the tabulated numerical values for zero line 
length and infinite fine length, for in some cases the general 
formulae assume indeterminate forms. 

Consider equation (24), for the load-end impedance correspond- 
ing to maximum efficiency. For I = 0, this equation assumes the 
form 







.2 _ 



2<) 2 P'=min. = 
J 1=0 



To evaluate this expression, differentiate both the numerator and 
the denominator of equation (24) with respect to I for a new 
numerator and a new denominator, respectively. Thus, 

„ aui cosh — Buz cos 

yf ! _ , 

aui cosh -f- Biii cos 
or 

ZoV= mio .= uv/^^- (30) 

J i=o V aui + Bvq 

Further, by substituting 

a = V% (zy — xb-jr rg) and B = V% (zy + xb — rg), 

and also 

_ ag + Bb Bg - ab 

Ul ~ g* + 6 2 ' ^ " g* + 6 2 ' 

which follow from the relation 

_ t /Z VZY V a+jB • . 

we obtain as the reduced form of (30), 

20 P'-min. = V -' (31) 

J 1=0 * g 



220 POWER RELATIONS IN LONG LINES 

This remarkably simple expression for Zo applies with con- 
siderable approximation to lines of considerable length, as shown 
by the slow variation of this quantity in the first three entries of 
Table XXIX. The above value of Zq is numerically equal to 
the load-end resistance required for a maximum efficiency of 
transmission with direct current over a leaky line of any length, 
for which see equation (34), Chapter II. 

136. Infinite Length Lines. — For a line of infinite length, 
equation (24) may be evaluated by dividing both the numerator 
and denominator by sinh 2 al. For infinite values of I, the fraction 

. , T vanishes, so that 
sinh 2 al 

Zo i»=min. = u. (32) 

J I-oo 

Therefore, with a very long line, to secure maximum efficiency, 
the load-end impedance should have a numerical value equal to 
u, the impedance of a fine of infinite length (see equation (13), 
Chapter VIII). 

For 1 = 0, equation (27) reduces to an indeterminate form. To 
evaluate, expand both numerator and denominator in series, re- 
taining only the first terms thereof. This yields 

■M.1-— -~ "'^t + ^M =0. (33) 

Thus, for short lines, the maximum efficiency is obtained when 
the load power-factor angle is small, and for the limiting case of 
zero length, the angle should be zero. 
To evaluate (27) f or I = oo , substitute 

sinh 2 2 al = 4 sinh 2 al cosh 2 al, 
sin 2 2 01 = 4 sin 2 01 cos 2 01, 

and divide both numerator and denominator by sinh 2 al. 

1+ _sin 2 Jtf_ 

sinh 2 al . 

sm<M = —uiy 2 —j= • (34) 

Ji,=max. / 

Uiyi coth 2 al + i^y 2 



J,=max. f • sm 2 01 cos 2 [ft 



sinh 4 aZ 

The fractions in both numerator and denominator vanish for 
infinite values of I, and coth 2 al becomes unity. Therefore 

Sm <fx> „=max. = 7= = — = - — • (35) 



GENERAL 221 

In this equation, — is equal to the sine of the angle, 0«, of the 

complex quantity U, which is the infinite-line impedance (see 
equation (13), Chapter VIII). Combining (32) and (35) we see 
that for very long (infinite) lines, the load-end impedance for 
maximum efficiency is 



Z ;=«, = U /~ d» = Mi — 
Ji;=max. 



M, (36) 



where U = y y = iti +juz = infinite line impedance. 

The imaginary component of Z in equation (36) is of the opposite 
algebraic sign from the imaginary component of U. This value of 
Z for maximum efficiency in a very long fine is not, as has often 
been assumed, that for which there is no reflected wave at the 
load end. From equations (10) and (10b), Chapter VI, the condi- 
tion necessary in order that the reflected wave be zero is that Z be 
equal to U, and this is not the same as the condition derived above. 
Therefore, for maximum efficiency, we can say that, in general, the 
load impedance must not be such that no reflected wave exists. 

137. General. — It is easily seen from equation (27) that, for 
maximum efficiency, the load power-factor angle, <£o, is always 
either positive or negative for a line of given fundamental con- 
stants, and never varies in algebraic sign as the length of the 
circuit is changed. The algebraic sign of <£o is always opposite 
to that of 1/2, and is thus always the same as that of u^. The angle 
of the load impedance, Z , being equal and of opposite sign to that 
of <fo, we can say that, in a line of any length, the load impedance 
for maximum efficiency must always have an angle of opposite 
sign to the angle of the infinite line impedance, U. In power 
transmission lines, where the angle of Y is usually greater than the 
angle of Z, this means that the angle of the load impedance for 
maximum efficiency is usually positive (the angle of U being 
negative). Summarized, for maximum efficiency we have: 

t n 

If - < - , lagging (inductive) load. 

If — = — (distortionless line), non-inductive or unity power 

factor load. 
L C 
If ->-, leading (condensive) load. 



222 POWER RELATIONS IN LONG LINES 

The above conditions depend only upon the line constants per 
unit length, and as before mentioned, are independent of the length 
of line involved. 

138. Proof that for Maximum Efficiency the Ratio between e 
and e Equals the Ratio between i and io> and that the Genera- 
tor Power-factor Angle is Equal in Magnitude but of Opposite 
Sign to the Load Power-factor Angle. — In section 133, some 
general conclusions were drawn from a number of numerical 
relations observed in a particular case. These relations may 
readily be established as general propositions. 

As indicated by equation (16), Chapter II, the solution for a 
line in which distance is counted positively from the generator 
towards the load is obtained by substituting — I for I in the equa- 
tions based on a positive direction counted from load towards 
generator. Following through the developments leading to the 
power equation (12) for alternating current circuits, we find the 
substitution of — I for I to still be permissible. The substitution 
is also justified in equation (24), where we find the value of Zq un- 
affected by the change — both numerator and denominator chang- 
ing sign. This establishes the fact that, for a minimum line loss 
with a given power Pand power-factor angle <f> (at the generator), 
the scalar value of z is equal to the scalar value of Zo. Therefore, 
e eo e i 

- = Z = 2o = -r, Or — = — , 

i v e<> to 

for maximum efficiency. 

Substituting z for Zo, <£ for fo, - for t\ and — I for I in (25), 

V 

77 = 1 + 2 sinh 2 al + 2 (1*22/2 + W1J/2 tan 4>) (sinh 2 al + sin 2 /3Z) 

-9-^if^ + -) sinh2aZ -(^ + -) sm2 ^i- 
& COS <p(\ z I \ z I ) 

1 



Comparing this equation with (25), we find the coefficient of 

cos <f> 

reversed in sign from the coefficient of — in equation (25). 

cos <po 

The coefficients of tan <f> and of tan <fo are alike in sign, so that, in 

equation (27), sin <f> will be of opposite algebraic sign to sin <fo. 

Such would not appear to be the case from an inspection of the 

equation for sin <fo alone, for the substitution of —I fori directly in 

(27) would not reverse the algebraic sign of sin fa. The double 



LOAD VOLTAGE FOR MAXIMUM ALL-DAY EFFICIENCY 223 

sign should be placed before the radical in equation (27) if + and 
— values of I are to be used. In the reduction from equation (25) 
to equation (26), we have for the last term, involving cos <f> 0) by 
substituting the equation for Zo, 



St • u o 7 • o on t Ai sinn 2 aZ — 7^2 sin 2 /3Z 

< (y x sinh 2 al - y 2 sin 2 0Z) V . uo . ^ u 

( ▼ Wi sinh 2 at + t<2 sin 2 /3t 




+ 



+ (wi sinh 2al — U2sin2 fit) v 

v V mi sinh 2 aZ — M2 sin 2 /3Z w 

The two radicals, which represent z , as well as u are essentially- 
positive, but in combining the expressions in parentheses to obtain 
the simplified form shown in (26), the algebraic sign of the entire 
term within the brackets is suppressed, if the positive sign only is 
retained before the resulting radical. Therefore, for maximum 
efficiency, <t> = — fo, and cos <f> = cos <j> . Then 



Since 
we have 



• ± 1 • ^ . 1 . 
ei cos <6 = - e<#o cos <6 , ei = - eoio. 

V V 



e eo 

z = - = Zq = —, 



i to 






to- 



The deductions drawn from the numerical results in section 
132 are thus proved to be true in general. 

139. Load Voltage for Maximum All-day Efficiency with 
Intermittent Load. — If a fine is to be operated so as to maintain 
constant load voltage and carries a given power load for a portion 
of the time and a negligible load for the remainder of the time (each 
day), the proper load-end voltage to ensure a minimum total daily 
loss in the line will be as determined in the following equations. 

In a given period of time, T, let t = the time during which full 
load is carried. Then T — t — time during which there is no load. 
Throughout the entire time, a constant e.m.f., e , is to be main- 
tained at the receiving end. By equation (13), the total energy 
loss in the line is 

W = t\P 2 sinh 2 al + § (e<?yi +i 2 Ui) sinh 2 al 
— h (eo 2 2/2 + itfvv) sin 2 /3Z 
+ 2 (Pom* + PJuiyt) (sinh 2 al + sin 2 fit) ) 
+ (T - (i «o 2 2/i sinh 2 al - \ efa sin 2 fl) . (37) 



224 POWER RELATIONS IN LONG LINES 

The problem is, with a given power-factor angle fo, and power 
Po, what should be the value of e in order that W be a, minimum 
when taken over a fixed time 77 

Placing ^ = k, 

and substituting, 

n i D 4. j. ' Po sec 00 

/V = "o tan fa, to = — • 

eo 

Then 

^r = A; P 2 sinh 2 al + ^ (eo 2 i/i + ^ Po 2 sec 2 <foui) sinh 2 al 

— 5 («o 2 2/2 + -3 Po 2 sec 2 ^0^2) sin 2 01 



+ 2 (P0U22/2 + Po tan 0oUi2/ 2 ) (sinh 2 al + sin 2 /3J) 

+ (1 - k) % fa sinh 2al-y 2 sin 2 /SO eo 2 . (38) 

v . W 

Let £j = variable portion of -~-. The value of eo for a minimum 

of p is to be found. 
From equation (38), 

p = eo 2 (t/i sinh 2al — y 2 sin 2 /3Z) 

+ \ (mi sinh 2 al - lit sin 2 00 fcP 2 sec 2 <fc>. (39) 
Co 

Differentiating with respect to eo, 

■j- = 2 eo (2/1 sinh 2 a£ — 1/2 sin 2 #) 
deo 

o 

1 (wi sinh 2 aZ - M2 sin 2 fit) kP 2 sec 2 <fc>. (40) 

eo 



Equating to zero to determine eo for a minimum p, 

'(ui sinh 2 a? — i*2 sin 2 01) kP 2 sec 2 <£o~l* 



«-[■ 



i/i sinh 2 ai — 2/2 sin 2 0Z 



(41) 



The above is the desired value of eo for which a minimum total 
energy loss will result. It is seen that the voltage necessary for 
such a condition varies as the fourth root of the fraction of time, 
k, during which the power P is to be used. Note that the product 
Po sec 0o is equal to the volt-amperes at the load. 



LINE LOSS WITH VARYING LOADS 225 

Comparing (41) with (24), we see that if z = load impedance 
for maximum efficiency as given by (24), we may write 

e = (z 2 fcPo 2 sec 2 <£„)*. (42) 

Taking up the illustrative numerical example in section 129, 
suppose k = 0.60 — that is, the 4000 kilowatts of power is to 
be used only six-tenths of the time. By the above equation, then, 
e = 47,736 volts, as compared with 54,238 volts for maximum 
efficiency when the power is utilized the entire time. 

The above development is for a very special type of loading, but 
in the following a method for determining the proper load voltage 
for any load distribution will be given. 

140. Line Loss with Varying Loads, and the Proper Load 
Voltage to Secure a Minimum of Such Loss. — Here, we again 
assume that the receiver, or load, voltage is to be maintained 
constant throughout all variations of load. The variable load 
must be given as a function of time, as by a load curve, for instance. 
We wish to determine what the constant receiver voltage should 
be in order that the total energy loss in the line, over the period of 
time in question, will be a minimum. 

If W represents the total energy loss during a given interval 
of time, by equation (27) and the transformations indicated in 
(38), during an interval of time, dt, 



dW = p 2 sinh 2 al + \ Uy, + ?£*?£** Ul ) sinh 2 al 

1/ . , P 2 sec 2 tf> \ . * 
- g ( e ° # 2 "• ~2 **» J sin 2 & l 

+ 2 (P M2?/2 + Po tan <^ Wi2/ 2 ) (sinh 2 al + sin 2 pi) \ dt. (43) 

Rearranging the terms and integrating between the limits 
and T, where T is the desired interval for which the total energy 
loss is to be a minimum, under the condition that eo remains 
constant, 

Podt 

o 

XT 
Pq tan 0o dt 

+ \ (yi sinh 2al-y 2 sin 2 fit) e 2 T 

1 C T 
+ \ (mi sinh 2 al - v* sin 2 pi) —„ J P 2 sec 2 tf> dt. (44) 



22(i 



POWER RELATIONS IN LONG LINES 



If, at all loads, eo is maintained constant, which is the condition 
for which equation (43) applies, we may differentiate this equation 
with respect to «o in order to determine the proper voltage for a 
minimum total line loss. 

dW 

—j— = (yi sinh 2 al — y 2 sin 2 pi) Teo 

1 C T 
. (mi sinh 2 al - u* sin 2 pi) I P 2 sec 2 fo dt. (45) 

6o Jo 



•A 

A 


*° 






























8 

i 






























66 


■- 






























80 


£ 


2 




























75 




X 
















[Po» 


Jcfy 


1 






70 


o 




























65 


1.00 






























60 


0.«»a 
























\ 






55 


n.'.X) 


11 










_y 


s 




Co 


>♦, 




\ 






50 


0.85 


10 








1/ 














H 


ts 




a 


0.80 


9 






/ 


















\\ 




40 


0.75 


8 


\ 


/ 














rp„ 






\ 




35 


0.70 


7 




























SO 




B 




























25 




5 




























*> 




4 




























15 




3 


























ID 




2 




























h 




1 








































T 


me 


Ho 


urs 


t'roi 


n M 


idn 


Kht 

















1 


{ 





8 l'O 1J2 1 


'?"?"■! 



Fig. 70. — Illustrative Load Curve for 24 Hours. 

Equating the above value of the first derivative to zero in order 
to solve for the value of e which will give a minimum energy loss, 

(mi sinh 2 al - Uz sin 2 pi) / P 2 sec 2 <£o dt f ,. as 
eo = Jo I • (4o) 

(yi sinh 2 al - y 2 sin 2 pi) T 



ILLUSTRATIVE EXAMPLE FROM DAILY LOAD CURVE 227 

By equation (24), Zq 2 may be substituted in the above for the 
fraction made up from the line constants, where Zq is the load-end 
impedance corresponding to maximum efficiency of transmission. 
Thus, 

eol = V7 (i f iV sec 2 <fc> dtf . (47) 

Po sec 0o = volt-amperes at the load, and since with a constant 
load voltage, the current is proportional to the number of volt- 
amperes, the value of the integral may be taken from charts 
produced either by a recording ammeter or a recording volt- 
ampere meter, after new curves, whose ordinates are equal to the 
squares of the ordinates of the original curves, are plotted. The 
integral between and T is equal to T times the average value of 
(volt-amperes) 2 . 

141. Illustrative Example from Daily Load Curve. — As an 
illustration of this formula, consider again the 400 mile line which 
has previously been used. See Table XXVI. Let this be -used 
to supply a load whose value and power-factor for each hour from 
midnight to midnight are as given in the following table. The 
quantity, P 2 sec 2 <f>o = (volt-amperes) 2 , is calculated for each entry 
and listed in the table. Figure 70 shows the power, power-fac- 
tor, and (volt-amperes) 2 curves as plotted from Table XXX. The 
value of the integral may be derived by means of a planimeter, 
or, since the hourly values are ordinarily connected by straight 
lines in the load curve, an average of the 24 ordinates will yield the 
area divided by the base, which is the quantity desired. 

From Table XXX, 

1 C T 
Average value of (P sec <ft)) 2 = ■=, J P 2 sec 2 <£<> dt 

= 62.8 X 10 12 (volt-amperes) 2 . 

From Table XXVII, using Zo for maximum efficiency = 666.53 
ohms, in equation (47), 

eo = V666T53 (62.8 X 10 12 )* - 72,680 volts. 

This, then, is the proper value of constant load voltage to be 
used if the total daily energy loss in the line is to be a minimum for 
the assumed load curve. 



228 



POWER RELATIONS IN LONG LINES 



TABLE XXX 

Data for Illustrative Daily Load, Power-factor, and (Volt- 
amperes)* Curves 
Time counted in hours from midnight 



( 


PoXio-* 


008*o 


Po'sec'^oXlO-" 





5.0 


0.80 


39.0 


1 


4.0 


0.75 


28.4 


2 


2.5 


0.75 


11.1 


3 


2.2 


0.80 


7.6 


4 


2.2 


0.80 


7.6 


5 


2.2 


0.80 


7.6 


6 


3.1 


0.85 


13.3 


7 


6.4 


0.85 


56.7 


8 


7.8 


0.90 


75.2 


9 


8.2 


0.90 


83.0 


10 


* 7.6 


0.92 


68.2 


11 


7.4 


0.90 


67.6 


12 


7.4 


0.90 


67.6 


13 


7.0 


0.92 


57.9 


14 


7.4 


0.93 


63.2 


15 


7.6 


0.95 


63.9 


16 


8.1 


0.96 


71.2 


17 


9.8 


0.97 


102.0 


18 


10.0 


0.97 


106.1 


19 


9.2 


0.95 


93.8 


20 


8.0 


0.90 


79.0 


21 


7.3 


0.90 


65.9 


22 


6.3 


0.87 


52.4 


23 


5.8 


0.85 


46.5 


T = 24 


1507.0 



142. Dependence of Proper Voltage upon the Leakage 
Coefficient. — The preceding discussions have been illustrated 
by a line whose dielectric loss coefficient, or conductance, g, is 
0.15 X 10 -6 mhos per mile. This corresponds to an energy loss 
of 500 watts per mile at a voltage to neutral of 57,700 volts, or a 
voltage between wires of a balanced three phase system of 100,000 
volts. Since this leakage coefficient is the only line constant of 
comparatively uncertain value, it may be of interest to ascertain 
the proper load end impedances for maximum efficiency, as well as 
the proper load end e.m.f . to give maximum all-day efficiency for 
the above load curve, when different values of g are assumed. 

Table XXXI contains values of the more important quantities 
entering into the determination of Zo for maximum efficiency by 
equation (24) and of the proper voltage eo by equation (47) for the 
load specified in Table XXX, when different values of g are used. 



DETERMINATION OF RESISTANCE AND LEAKAGE LOSSES 229 

TABLE XXXI 

Line Constants and Conditions for Maximum Efficiency in a 
400 Mile Line with Different Values of g 



gXW 


0.00 


0.05 


0.10 


0.15 


0.20 


0.25 


0.30 


2al 


0.28981 


0.29741 


0.30501 


0.31262 


0.32022 


0.32784 


0.33545 


2# 


95.752 


95.682 


95.608 


95.538 


95.468 


95.402 


95.336 


Ui 


379.54 


379.83 


380.10 


380.34 


380.56 


380.75 


380.92 


«2 


-65.819 


-64.093 


-62.365 


-60.630 


-58.897 


-57.161 


-55.420 


z% 


755.25 


721.51 


692.23 


666.53 


643.79 


623.46 


605.29 


e 


77,360 


75,610 


74,060 


72,680 


71,430 


70,290 


69,260 



The above values are shown plotted in Fig. 71. 





750 








































«0 

X 


«| 


2al 






































io 8 


710 


0.35 






































78 


720 








































77 




.34 








V 






























76 


700 








































75 




jn 








s" 






























74 


680 




























€k 












73 




.:« 






































W 


GOO 








































71 




ja 






































70 


010 






























% 










69 




.••so 






































68 


0-20 




























\Zfl 












67 




.2!) 






































66 


000 








































i 











































0.04 



0.28 



0.32 



0.08 0.12 0.16 0.20 0.24 
Dielectric Loss Coefficient, g, x 10 8 

Fig. 71. — Conditions for Maximum All-day Efficiency for the Load Curve of 

Fig. 70, with Different Values of Leakage g. 

143. Determination of the Resistance and Leakage Losses 
Separately; Graphical Method. — Equation (13) gives the total 
power lost in the line, which, of course, is made up of the sum of 
the i?r and e 2 g losses for every element of the line length. As it 
may sometimes be desirable to compute the losses due to each 
cause separately, formulae for this purpose will be given here. 



230 POWER RELATIONS IN LONG LINES 

Two methods are available — graphical and analytical — the 
latter usually being preferable on account of the greater ease in 
application and the higher accuracy of the results. 

To determine the resistance loss for a given load by the graphical 
method, calculate the line current for a number of equidistant 
points along the line and plot, in polar coordinates, the curve 
representing i as a function of line length. The radius, R, in the 
polar curve is drawn to such a scale as to represent the line current. 
Suppose a unit angle in our diagram represents h units of line 
length — angles being counted in radians and distances along the 
line in whatever units we may select. Let a unit length of radius 
represent C amperes of current. Then, in the diagram, for any 
point in the line distant I from the load 

R = 1 i = CR and 6 = ^ , I = hB, (48) 

which are the relations between the coordinates of a point on the 
polar curve and the quantities in the line. The total resistance 
loss in the line is 



= r Ppdl, 

t/0 



(49) 



where r is the resistance of the line per unit length (the same unit 
of length as used in defining h). 
The area of the polar curve is 

R? 



-i 



2 dd. (50) 



If we substitute (48) in (49), the integration between limits zero 
and I is replaced by integration between limits and 0, and by 
this substitution, 

P r = rhC 2 f 6 R 2 M. 



f 6 R 2 dd. (51) 



By equation (50), however, the integral in the above equation 
represents twice the area of the curve, 2 A. Therefore, 

P r = resistance loss = 2 r^C 2 A. (52) 

In a similar manner, the total leakage loss may be determined 
by plotting a new polar curve representing the electromotive force 
at all points along the line. If C e is the number of volts repre- 



ILLUSTRATION OF GRAPHICAL METHOD 231 

sented by a radius of unit length in the diagram, h is the number of 
units of line length per radian in the diagram, and A e is the area 
of the new polar curve, 

P, = 2ghCM., (53) 

where g is the conductance of the line per unit length. 

144. Illustration of Graphical Method. — As an example, the 
power losses in the line whose voltage and current distribution are 
as given in Table XIII may be calculated. In the polar diagrams, 




1 2 

Fig. 72. — Graphical Determination of Separate Power Losses. 

or curves, let one degree of angle represent ten miles of line. 
Then 

h = !?? 10 = 572.96. 

Let one inch in the diagram of currents represent 40 amperes, 
and in the voltage diagram, let one inch represent 20,000 volts. 
Then C = 40 and C, = 20,000. For this line, r - 0.275 ohm 
per mile and g = 0.15 X 10 -6 mho per mile. Fig. 72 shows 
the current and voltage diagrams as plotted from the data in 
Table XIII. From these figures, by the aid of a planimeter, 

A = 15.13 square inches, and A e = 10.52 square inches. 



232 POWER RELATIONS IN LONG LINES 

Therefore, 

P r = 2 X 0.275 X 572.96 X (40) 2 X 15.13 - 7,629,000 watts. 

P„ = 2 X 0.15 X 10-« X 572.96 X (20,000) 2 X 10.52 = 723,000 watts. 

P' = P r + P g = total line loss = 8,352,000 watts. 

The above value of P' added to P should equal the total power 
at the end of the 2000 miles. Forming the sum, we obtain 
9,482,000 watts, which differs by 1.4 per cent from the value given 
in Table XIII. Considering that the quantities are all obtained 
graphically, the discrepancy is no more than might be expected. 
Nevertheless, a much closer result is obtained for the mid-point 
of the line. For this length, the areas of the current and e.m.f. 
curves are, respectively, 6.27 and 3.12 sq. in. Computing the 
separate losses and adding to P , we obtain 4,510,000 watts for 
the power at a distance of 1000 miles. The error here is less than 
one per cent. 

At best, the graphical process is difficult to apply on account 
of the necessity of determining the current and e.m.f. at a sufficient 
number of points along the line to enable smooth curves to be 
drawn. 

145. Separate Losses Analytically. — To determine the re- 
sistance loss analytically, we must form the integral 



\ = r fi^dl, (54) 

Jo 



and this necessitates a knowledge of the square of the scalar value 
of current at any point in the line. At any point, 

I = ii+ ji*, (55) 

and in equation (5) we have given the values of t'i and U in terms 
of io r and V, the two components of the load current. Since the 
two components of I are at right angles to each other 

* = h 1 + k". (56) 

Equation (54) is then 

Pr==r X <* + *&* (57) 



SEPARATE LOSSES ANALYTICALLY 233 

Substituting (5) in (57), after some transformation, 



P, 



■ f I < t 2 cosh 2 al cos 2 (51 + i 2 sinh 2 al sin 2 j8Z 

«/o ( 

+ e 2 -5 sinh 2 aZ cos 2 01 + e 2 -; cosh 2 al sin 2 /3Z 
it 2 w 2 



+ 2 io' (eoVi — «o"2/2) cosh al sinh aZ 
+ 2 z'o" (eo'i/2 + e "t/i) cosh al sinh aZ 
— 2 V (e</2/2 + e "t/i) cos fil sin /3Z 

+ 2 f " («b'yi - «o"?/2) cos /3Z sin /3Z > dl. (58) 

Introducing the relations, 

Po = e W + e "io" and P >' = eo%" - eo'V, 

and combining terms, 

P T = r f l j^o 2 + e 2 ^)sinh 2 aZ 

+ i 2 cos 2 j3Z + e 2 -5 sin 2 flZ 
IF 

+ (P yi + P '' 2/2) sinh 2 aZ 
- (Po^-Po'Vi) sin 2 0ZJdZ. 

1 y 

Where -5 = yi 2 + 1/2 2 = scalar value of „• 

It- ^ 

Integrating, 

+ 5- (PoVi + Po'y«) cosh 2 al 



(59) 



+ -^ (P0I/2 - Po J 2/i) cos 2 0ZT- (60) 



234 POWER RELATIONS IN LONG LINES 

Subtracting the value at the lower limit from the value at the 
upper limit, the final form becomes 






+ ^(Po2/i + iV'i/2)sinh 2 aJ 

- 1 (Pw ~ Po'Vi) sin 2 fil I watts. (61) 

The equation for the total conductance loss may be written by 
symmetry from equation (61), with the exception that the algebraic 
signs before the terms containing P ' must be reversed. This 
arises from the fact that, in squaring the expressions for e x and e% 
in equation (4), products of the form to'eo" — «o"e</ replace the 
products of the form e</to" — eo'%' as obtained in the above de- 
velopment for P„ 
Thus, 



-£ 



e 2 dl = g \ - — (eo 2 + ioV) sinh 2 al 
(4a 

+ -^(eo 2 -io 2 u 2 )sin2# 

+ - (PqUi — Po'Uz) sinh 2 al 
a 

- | (P0M2 + Po'mi) sin 2 01 1 watts. (62) 

Equations (61) and (62) present the desired results. Though 
they give the losses separately, it must not be assumed that each 
of these losses is the same as the total line loss which would occur 
if the other were absent, for in the event of the conductance loss, 
say, being zero by virtue of a zero value of g, the value of P r by 
equation (61) would not be the same as for a condition of g being 
finite because of the change of a, /3, u, y h y 2 , etc., with the change 
in g. To determine P T when g = 0, all quantities entering the 
equation for P, must be determined for this value of g. This fact 
seems to have been overlooked by a number of writers who have 
developed formulae for power loss in lines. 



SEPARATE LOSSES AT NO LOAD 235 

148. Illustration of Analytical Method. — As an illustration 
of these equations, let us calculate the separate losses for the line 
whose constants are given in Table XXVI, when this line supplies 
a load of 6000 kilowatts under maximum efficiency conditions. 
By Table XXVII, Zq = 666.53 ohms, and fa = -40.938 degrees, 
for maximum efficiency. e and t are then computed by equation 
(21), and Po 1 by the equation 

Po' = Po tan </>o. 

The main features of the computation appear below in Table 
XXXII. 

TABLE XXXII 
Calculation of Separate Line Losses 
Equations (61) and (62) 

Po 6,000,000 watts 

Po> -5,204,200 volt-amperes 

<j>o -40.938 degrees 
e 72,760 volts 
io 109 . 162 amperes 

For the four terms within the brackets 

Resistance loss Conductance loss 

No. 1 9,677,000 1,435,430 X 10 6 

2 -2,838,190 421,000 X 10 6 

3 835,660 123,960 X 10 6 

4 -4,155,100 616,340 X 10 6 

P r /r 3,519,370 P g /g 2,596,730 X 10 6 

P r 967,870 P g 389,510 watts 

P = P r + Pg - 1,357,380 watts 
P = Po + P' = 7,357,380 watts 

9 -Eff. = ^ = 0.81551 

The above agrees with the value of maximum efficiency given 
in Table XXVIII as obtained from the efficiency equation (26). 

If we compute the resistance loss by the square of the load 
current times the total line resistance, 1,310,000 watts is the 
result. The square of the load voltage times the total line con- 
ductance gives 317,000 watts. The error in the approximation by 
this method is considerable. 

147. Separate Losses at No Load. — Under no-load conditions, 
P = 0, and P ' = 0. Equations (61) and (62) then become very 
simple. 



236 



POWER RELATIONS IN LONG LINES 



Short Circuit 

sin 2 0ly 



p "I r / sinh 2 al sin2#\ . 

1 = |/sjnh^_sJn2^A 



Open Circuit 



p ] r f sinh 2 al sjn2#\ 1 2 

Ji =o 4\ a / 



(63) 



(64) 



Series expansions for the above formulae may be easily obtained, 
but no advantage would be gained — the above forms lending 
themselves admirably to the requirements of the computer. 



CHAPTER XI 

VOLTAGE REGULATION 

Case I. Both Components of Load Become Zero 
Simultaneously 

148. Preliminary Developments. — Since most power genera- 
tion takes place under practically constant voltage conditions, 
and further, since the major portion of electrical apparatus requires 
essentially a constant voltage, regard must be taken of the fluctua- 
tions of the load-end e.m.f . of a line when the load varies in magni- 
tude. Adjustment as well as regulation of load-end voltage is 
often accomplished by means of over or under excited synchronous 
motors, which thereby consume large currents in quadrature with 
the impressed e.m.f. It becomes necessary, then, to be able to 
predetermine the magnitudes of such currents required to produce 
the desired effect. 

For any given receiver load, the voltage regulation of a trans- 
mission line is denned as, 

-r, . x . Rise in voltage from load to no load ,„ N 

Regulation = Load voltage (1) 

the generator e.m.f. being maintained constant throughout the 
change. 

At a given load J , and load-end voltage E , the generator 
voltage is 

E = E cosh VI + I U sinh VI, 

and if this generator e.m.f. be maintained constant while the load, 
and therefore 7 , is reduced to zero, the new value of load-end 
voltage becomes, by equation (8), Chapter IX, 

E ' = E sech VI = E + I U tanh VI. (2) 

This equation may be written, 

E = E ' - hU tanh VI, 
237 



238 VOLTAGE REGULATION 

a form which could have been written immediately from a gen- 
eralization discussed in a following section. 
The difference between the two load-end voltages is 

E ' - E Q = hU tanh VI. (3) 

149. Scalar Values. — This difference divided by E does not 
give the regulation as defined in equation (1), for the definition 
is based upon the scalar values of the electromotive forces under 
the two conditions, while in (3) a vector difference is obtained. 
Using equation (3) in (1) would yield a complex expression, or 
value, for the regulation. The numerical value of this complex 
result would, in general, be greater than the true regulation, since 
the scalar value of the vector difference between two quantities 
is greater than the difference between the scalar values of the 
quantities themselves (unless the two quantities are in phase, 
when the results coincide). The expressions for line regulation 
must, then, be based upon scalar values. 

150. Introduction of General Form of Equations. — The 
transmission line equations appear in the form, see equation (31), 
Chapter VI, 

E = AE + Bio, S - (fc + 3<h) E + (h + j&,) I , 
I-AI f + CE , I - (d + j(h) h + (ex + fa) Eo, 

A = cosh VI, B=U sinh VI, C = jj sinh VI. 

Under no load, 

E e 2 e 2 

Eo' = ; — : — , «o' 2 = — r~ i o = -v (5) 

But E = AE + Bio, so that if 

Eo = e' + je", and I = i' + ji", 

, 2 _ {aie' - a^e" + bii' - H"Y + {axe" + o 2 e / + bd" + W ,„, 
eo - - 2 (pj 

By squaring, as indicated, and subtracting eo 2 = e' 2 ■+• e" 2 , 

eo'» - eo 2 = \ [bft' 2 + hH m + bH m + Wi' 2 
a 1 

+ 2 (—diOze'e" + aibie'i' — aib&'i" — ajb\e"i' 

+ a&e"i" - b x \hi'i" + a x a*Je" + aM'i" 

+ a x b&"i' + aibxe'i" + (hhe'i' + bJhi'i")]. (7) 



(4) 



CASE I 239 

Combining terms, and replacing by equivalents (see sections 122 
and 123), 

eo' 2 - e 2 = \ \bW + 2 (<fa&i + 0262) P + 2 (0261 - afo) P Q 3 '\ . (8) 



Referring again to the form of notation described in Chapter 
X, section 122, where products of the form U1V1 -f- 1^2 are written 
U • V, and products of the form U1V2 — utf) \ are written U X V, 
a simplification in the equations may be made. 

Thus, equation (8) becomes 

eo'2 -eo 2 = \ \b*i<> 2 + 2A-BP -2Ax BP '\. (8a) 

- 151. Reactive Volt-Amperes Required at Load for Zero Regu- 
lation. — The above equation gives the difference between the 
square of the scalar value of the no-load voltage, eo' 2 , and the square 
of the scalar value of the load voltage, e 2 . If it is desired to so 
operate the system that the regulation as defined in equation (1) 
be zero, the amount of reactive power, or volt-amperes, Pq\ 
required to attain such a condition may be determined from equa- 
tion (8) by placing e ' 2 — eo 2 = 0. The voltage at no load will 
then coincide with the load voltage, numerically. i Q 2 may be 
eliminated from the equation by the relation 

eoto = Vp * + P >\ ; 2 = P ° 2 t 2 P ^ ' (9) 

Making this substitution, and solving the resulting quadratic 
in /V, 

fV = rj^ }ai&2— <kh 

± V (ai&2-a2&i) 2 - ^[^o 2 +2 (axk+a^Pol j (10) 

or in simplified form 

PJ = |? j AxB±\f (AxB) 2 - ^ 2 |"^ Po 2 +2A. J BPo]| . (10a) 



240 



VOLTAGE REGULATION 



In the above equation, 

d = cosh al cos /SZ, Oz = sinh al sin 01, 
61 = Ui sinh al cos fil — v* cosh aZ sin /3Z, 
6j = Ui cosh aZ sin /3Z + Ut sinh aZ cos /3Z, 
6» = 61 2 + W. 



(ID 



From these values, the combinations which enter in the equation 
for P ' may be written. Thus, 

A • B = aibi + a«&2 = U\ cosh aZ sinh al — Ui cos /3Z sin /3Z, 
i X B = 0162 — a 2 6i = ii2 cosh aZ sinh al + Mi cos /3Z sin /3Z, (12) 

6 2 = m 2 (sinh 2 aZ + sin 2 01). 

In computing, values of the combinations only as given by the 
above formulae need be determined, and the labor involved is no 
greater than that required to determine a h a-t, 61, 62, and b 2 sepa- 
rately. 

As may, at times, be found desirable, equation (10) can be 
expressed in terms of the components of current in phase with and 
in quadrature with the load voltage eo. Thus, 



io p = in phase, or power, component = — , 

P } 
in' = quadrature component = — -• 

£o 



(13) 



As seen from an inspection of equation (10), two values of P ' 
or of io 7 are obtained for which zero regulation is secured. For 
power loads greater than a certain definite amount, depending 
upon the line properties and the load voltage eo, the quantity 
within the radical becomes negative, and therefore P J and itf 
become imaginary, which means physically that, under such con- 
ditions of line and load, it is impossible to secure zero voltage 
regulation by adding quadrature components of current to the 
load. 

152. Conditions for Minimum Voltage Regulation. — Al- 
though under such cases as above it is impossible to secure zero 
regulation, the quadrature component of the load current may be 
so adjusted as to secure a minimum voltage regulation. The 
substitution of (9) in (8) gives 



* 



'2 _ 



eo 



*=I 



^ + 2A.BP + V^-2A xBPod , 
eo 2 e z ) 



(14) 



CASE I 241 

from which the change in voltage may readily be computed. To 
make this change, and therefore the voltage regulation, a minimum, 
differentiate with respect to Po 1 and solve for P ' after equating 
the first derivative to zero. 



d (e</ 2 - e 2 ) 



dP > 
Solving the above, after equating to zero, 



-*[*%* -* A * B ] (15) 



W2 sinh 2 al + Ui sin 2 01 , 
e 



P i] = AXB 2 = 

Jreg.=min. & 2 " 2 u? (sinh 2 al + sin 2 01) 

]_ AxB _ Uz sinh 2al-{-Ui sin 2 ft 
reg.=min. " "b^ eo == 2 u 2 (sinh 2 ai + sin 2 ftZ) 



(16) 



For power loads below the critical value above which zero regula- 
tion cannot be secured, the value of Po 1 ' determined by the above 
formula will give a negative regulation — that is, the load voltage 
will be greater than the no-load voltage. 

153. Conditions for any Voltage Regulation. — If we so 
desire, the quadrature component of load current may be so 
adjusted as to give any arbitrarily selected value to the voltage 
regulation. Let it be required to determine Po 1 ' to give a specified 
regulation, R. Then, if e = load voltage and e</ = no-load 
voltage, 

R = f°l^i°, eo ' m (1 + B) *. (17) 

Co 
The left hand member of (14) becomes 

eo '2-e 2 = (2R + R*)e 2 . (18) 

Introducing this value, and solving the quadratic in P } , 



± 



y (Ax£) 2 -|^ 2 Po 2 +2A .£P -a 2 (2fl+# 2 )eo 2 l j- ( 19 ) 



This equation, like (10), may give imaginary values of Po 1 . With 
given line properties and load voltage e , real values of Po 1 will be 
secured through a greater range in P as independent variable 
when finite positive values of R are substituted in (19) than could 



242 VOLTAGE REGULATION 

be obtained from (10). This should be obvious — it really 
amounts to saying that a larger amount of power can be trans- 
mitted with poor voltage regulation than with good. Again, two 
values of P ', for which the regulation has its specified value, are 
obtained from the equation. 

164. Solutions in Terms of Load Admittance. — If the load 
admittance Y is given by 

Y = g + jb , then P = e o 2 0o, and P y = e 2 b . (20) 

It is often desirable to set up a solution in terms of 6 and g rather 
than in terms of e , Po, and P ' for then the solution is independent 
of the actual voltage employed and of the power transmitted. 
Substituting (20) in (19), 

b =^\AxB±V(AxBy-b 2 [b 2 g 2 +2A-Bg -a i (2R+R*)]}-(21) 

Placing R = 0, we get the special case given by equation (10), 

&ol = \ 2 { A X B ± V(A X B) 2 - &V - 2 b*A • Bg \. (22) 

The above values of g and 6 obtain under load, and must then 
become zero simultaneously, according to the bases for the develop- 
ment of the above equations. With constant generator voltage 
and a variable load Po, a variation of P J according to equation 
(10) will maintain constant load voltage — Po' disappearing 
simultaneously with P , but not being proportional thereto for 
intermediate loads. 

155. Numerical Illustration. — As illustration, consider again 
the 400 mile, 60 cycle, power line described in Table XXVI. 
Using the line constants given in the various foregoing tables, 
equation (21) becomes numerically, 



10 3 6 = 2.1141±V4.4693-10 6 ^o 2 -2132.4sr +5.6057(2 J R+« 2 ).(23) 

From this equation the values of 6 given in the following table 
were obtained. Both values of 6 corresponding to particular 
values of R and gfo are tabulated. Positive 6 indicates a con- 
densive (leading) load, and negative b an inductive (lagging) load 
component. 



CASE I 



243 



TABLE XXXIII 

Values of b = Load Susceptance Required to give a Regulation R with 
a Load Conductance go, when Both g and b Become Zero at No 
Load. R Defined by Equation (1), 6 X 10 3 Tabulated 








0.0002 


0.0004 


0.0006 


0.0008 


0.0010 


0.0012 


0.0014 


0.0016 


0.0018 





4.228 


4.115 


3.973 


3.796 


3.571 


3.270 


2.800 


Imag. 


Imag. 


Imag. 





0.113 


0.255 


0.432 


0.657 


0.958 


1.428 


Imag. 


Imag. 


Imag. 


0.04 


4.334 


4.226 


4.092 


3.927 


3.721 


3.454 


3.077 


Imag. 


Imag. 


Imag. 


-0.106 


0.002 


0.136 


0.301 


0.508 


0.775 


1.151 


Imag. 


Imag. 


Imag. 


0.08 


4.438 


4.336 


4.209 


4.054 


3.862 


3.621 


3.299 


2.790 


Imag. 


Imag. 


-0.210 


-0.108 


0.019 


0.174 


0.366 


0.608 


0.930 


1.438 


Imag. 


Imag. 


0.12 


4.542 


4.444 


4.324 


4.177 


3.998 


3.776 


3.491 


3.089 


Imag. 


Imag. 


-0.314 


-0.216 


-0.096 


0.051 


0.230 


0.452 


0.737 


1.139 


Imag. 


Imag. 


0.16 


4.645 


4.551 


4.436 


4.298 


4.129 


3.924 


3.666 


3.323 


2.774 


Imag. 


-0.417 


-0.323 


-0.208 


-0.069 


0.099 


0.305 


0.562 


0.905 


1.455 


Imag. 


0.20 


4.748 


4.658 


4.548 


4.416 


4.256 


4.064 


3.838 


3.525 


3.096 


Imag. 


-0.520 


-0.430 


-0.320 


^-0.187 


-0.028 


0.164 


0.400 


0.703 


1.132 


Imag. 



In Fig. 73 the values of b given in the foregoing table are plotted 
as functions of g . Multiplication of go and b by e 2 gives P and 
Po', respectively. For example, if e = 80,000 volts, and a load of 
7680 kilowatts is to be supplied, g = 0.0012, and for a regulation 
of 8 per cent = 0.08, from the tabular data or curves, 6 = 0.003299 
or 0.000930. Jtf then is 21,130 or 5950 kilovolt-amperes. The 
smaller is the only practical value to use, so the final load power- 
factor is 



cos <t>o = P.F. = cos tan -1 



5950 
7680 



= 0.79, leading. 



166. Nature of Curves Relating go and b . — The curves in 
Fig. 73 are ellipses, with their common center displaced to the 

left of the Y axis, and above the X axis by an amount — — r^ * 

A X B 

— — jn — . The standard form for the equation of an ellipse may 

be derived easily from equation (21) by placing A X B in the left 
hand member and then squaring. Proceeding in this way, we 
obtain as the final form, 



244 



VOLTAGE REGULATION 



The curves in Fig. 73 plotted from equation (24) are ellipses with 
unequal axes only because the scales used in plotting g and b are 
different. For equal scales, the curves become concentric circles, 
for (24) is the equation of a circle whose radius is 



Radius = £ (1 + R) - 



scalar value of cosh VI 
scalar value of U sinh VI 



(1 + 22). (25) 



6, 


































10 


































4.8 
4.4 
4.0 
3.B 
3.2 
2.8 
2.4 














































































Rer. 




























^20 







































































































hi. 


for 


iBi 


inn 


in 1 


tl'LT. 


=0( 


o-ui 


41 


] 1 


\ 


J. 


u 

1.0 

u 

0.8 
I 




































































































































0.4 
0.8 
1.8 










































































Co* 


10' 
















a 





2 





4 





(3 





8 


1 





1 


2 


1 


1 


1JS 







































































































Fig. 73. — Load Susceptance b Required to Give a Regulation R 



The coordinates of the center of the system of circles, with refer- 
ence to the origin for g and 6 , are 



A • B _ aibi + Q2&2 _ 1 Ui sinh 2 al — ity sin 2 /3Z 
6 2 ^ = ~2 m 2 (sinh 2 oi + sin 2 01) ' 



and 



Ax5 aJh — 0261 1 W2 sinh 2al-\- Ui sin 2 /SZ 



b 2 



2 u 2 (sinh 2 «Z + sin 2 #) 



(26) 



CASE I 245 

157. Geometrical Construction of Curves. — From the in- 
formation in equations (25) and (26), the entire system of curves 
may be drawn very quickly with the aid of a compass. First, 
locate the common center of the circles with respect to the origin 
of coordinates, and then draw in the circles with radii determined 
by equation (25), according to the values of regulation, R, desired. 
For any given load, g and b , the regulation may be ascertained 
from the chart by measuring the distance from the center of the 
system of circles to the point determined by g and 6 . This 

length divided by j- gives (1 + R), and thus R. For the par- 
ticular problem in hand, the coordinates of the center of the circles 
are, by (26), 

• -0.0010661 and +0.0021141. 

By (25), for the radii of the circles we have 

Radius = 0.0023677 (1 + R) 

= distance from center of circles to origin +0.0023677 R. 

Figure 74 shows the family of circles drawn according to the 
above data. Reading from the center outwards, the curves are 
for 0, 0.04, 0.08, 0.12, 0.16, 0.20, and 0.24, values of regulation R. 
The curves are not numbered, for to do so would crowd the dia- 
gram unduly. The data from this set of curves should duplicate 
that given in Table XXXIII and plotted in Fig. 73. 

The power-factor angle of the load may be found graphically 
from the chart by taking the angle between the g axis and the 
line joining the point (g , b ) with the origin. Obviously, the scale 
of the diagram may be so changed as to read in amperes, or in 
kilowatts and kilovolt-amperes, for any particular value of e Q . 
Attention may here be called to the fact that this method of curve 
construction does not apply to transmission line problems only, 
but to any system for which linear equations of the form 

E = AE + Bio 

apply. Since the general electric circuit is represented by such 
an equation, the scheme applies thereto. Analytically, then, 
electric circuits are different only in the manner in which the 
coefficients A and B are different, in the above type form of 
equation. As before pointed out, the solution for an electrical 



240 



VOLTAGE REGULATION 



system amounts to the determination of these constants. It is to 
be noted in equation (25) that the coefficient of (1 + R) is the 
reciprocal of the scalar value of the short-circuit impedance. 













































*,* 


w 
































4.8 
































































1.0 
































































3.2 




























! 
































i 
j 

1 




2.4 








111 














_ 




tor 


i 


































l.fi 








// 
























































(1.8 
































































i) 




% 








V 


10 


■1 








-1 


fi 


■4 


a 





•1 


v& 




1 


.0 


2 


4 


■■>. 


2 














0.8 
































































-1.0 






Item 


Bag 


fron 


CCIlt 


cr m 
1'', n 


tWM 


• '. 




















and 


21 pi 


r M 


t ri't. 


ulut 


cm. 





































Fig. 74. — Graphical Method for Plotting the Curves Relating 
ffo and 6 for Any Value of i2. 

168. Relations between Vector Diagrams and Foregoing 
Curves. — The vector diagram from which data may be taken 
for the plotting of such curves as Fig. 73 or Fig. 74 is of interest, 
since it brings out in a clear manner the significance of the graphical 
construction just described. Fig. 75 is this vector diagram. 

If eo = load voltage, e ' = (1 + R) «o *= receiver voltage at no 
load, with regulation R. Since the scalar value of e (at the 
generator) remains constant, and 

e = a€o' = a (1 + R) e , 

we see that the scalar value of e to produce a regulation R must be 
given by the above equation. That is, under load, the end of the 



CASE I 



247 



vector E must lie on a circle drawn about the origin with a radius 
a (1 + R) eo. In Fig. 75, the vector AE is shown, with the family 
of circles for different values of R. 

To the vector AE are added values of BgoE , for various values 
of g , thus completing the vectorial representation of the equation 

E = AE + BI , 

for values of I = (goE ) in phase with the load e.m.f. This vector 
addition gives the points spaced along the line AEqM, correspond- 




A =0.69020 19I69S. 
BsJ89L61,l72lM, 
SeeiableuXIV 

Fig. 75. — Vector Diagram for Regulation Calculations. See Fig. 74. 

ing to different values of g . A wattless component of load current 
jboEo requires the addition of another vector jBb E to the vector 
from the origin to the point on the line AE M, located as de scrib ed. 
jBboEo is at right angles to BgoE , and the refore to the line AEqM. 
Drawing a line through the point on AEoM which corresponds to 
some particular value of g , we determine the numerical value of 
the quantity BboEo required to give a regulation R by the per- 
pendicular distance from the line AE M to the ntersections of 
the new line with the proper circle. These perpendicular distances 
are the intercepts on the new line, NQ, between the line AEqM 
and the intersections of NQ with the circles. These intercepts 
divided by beo give the required values of bo. In the diagram, 
a number of lines parallel to NQ are drawn, each one corresponding 



248 VOLTAGE REGULATION 

to a particular value of g . By drawing the diagram to be Q times 
the voltage scale, distances along the line AEqM represent g 
directly, and distances along the line NQ represent 6 . The 
mechanical construction of the curves in Fig. 74 is accomplished 
by this method. If the above mentioned scale is used, distances 
along AEqM correspond to distances along the g axis in Fig. 74. 
The point AE Q in Fig. 75 is the origin in Fig. 74, and distances 
along the lines NQ correspond to distances along the & axis. The 
6o axis itself is r epre sented by the line parallel to NQ drawn 
through the point AE . The origin in Fig. 75 corresponds to the 
center of the circles in Fig. 74. 

If coordinate paper is used in drawing the curves, as in Fig. 74, 
the desired data is immediately obtained, without the necessity 
of actually constructing the vector diagram, Fig. 75. 

159. General. — In the above developments it has been 
assumed that the real power P and the wattless volt-amperes 
Po' both become zero at no load, in determining the rise in voltage 
at the receiving end from load to no-load conditions. When such 
is the case, P</ may, as has been shown, be so adjusted as to give 
zero voltage regulation, according to the definition in (1). If the 
reactive volt-amperes, Po', are consumed by a separate machine, 
or machines, it may happen that P and P J would not both become 
zero, but only the real power. Under such conditions, a different 
value of Po* is required to give a specified voltage regulation. If 
we assume that P ' remains constant while the real power, P , 
becomes zero, a value of P J can be determined for which the 
regulation will have a certain specified value. Actually, P ' is 
made up of two parts, one of which arises from the reactive volt- 
amperes provided to the load proper, and which, therefore, be- 
comes zero simultaneously with P , and the other which consists 
of the wattless volt-amperes supplied to whatever voltage regulat- 
ing device (as a synchronous condenser) is permanently connected 
to the load end of the line, and which, therefore, does not become 
zero simultaneously with P . If we consider the power required 
to drive the synchronous condenser, the situation becomes still 
more complicated. When such a condenser is used to consume a 
wattless component of current, a condition of constant reactive 
volt-amperes consumed thereby is not secured, for on removing 
the load, the voltage rises, and on account of this increased voltage, 
the synchronous condenser will consume a smaller wattless com- 



CASE II 249 

ponent of current, if leading, and a larger wattless component, if 
lagging. The portion of Pn 1 ' supplied to the synchronous condenser 
is not constant with varying voltage; neither is the wattless com- 
ponent of current consumed thereby, nor the equivalent suscept- 
ance representing the same. To express, analytically, the per- 
formance characteristics of such synchronous condensers, and to 
incorporate these expressions in solutions for line regulation, is 
scarcely practicable, but in the following, a number of solutions, 
based upon different assumed load-end conditions, are given. 

Case I, for P and P ' disappearing simultaneously, has. just 
been given. 

Case II. Total Load-end Susceptance Constant 

160. Load-end Voltages in Terms of Load-end Admittances. 
— Here we assume that the total load-end susceptance, 6 , 
remains constant while the load-end conductance, g , becomes zero 
at no load. 

Let 

Y = go + i&o = admittance under load. 1 

Yq'= -\- jb = admittance at no load. J 
By equation (20), Chapter II, in general, for generator voltage E, 

E ° - coshVl + WsinhVl ~ l0ad v0ltage > (28) 

and since 

cosh VI = a\ + j(h and U sinh VI = 6i + j'62, 

we have, under load, 

E 



En = 



'° «i + j(h + (go + jb ) (61 + jb 2 ) 
and at no load (g = 0), 

Eh = 



di + j<h + jb (61 + jb 2 ) 



(29) 



Let R = voltage regulation, with constant generator voltage, e. 
Then e ' = (1 + R) e , and by substituting (29), 

e 2 (l + RYe* 

60 — 



(a t - W>2) 2 + (02 + W>i) 2 (ai + 0o&i - 6 6 2 ) 2 + (02 + g &2 + W>i) 2 
= (l + fl) 2 eo 2 . (30) 



250 VOLTAGE REGULATION 

Clearing of fractions, combining terms, and substituting: 

fll 2 + 0^ = 0* and bf + k 2 = 6 2 , 
we obtain 

- 6* (2 R + «*) &o 2 + 2 (2 R + # 2 ) (o,6, - aA) &o 

+ 2 (0,6! + 0262) f/o + 6V - a 2 (2/2 + fl») = 0, 

or, when written in simplified notation, 

- 6 1 (2 # + A 2 ) 60 2 + 2 (2 # + R?) A X B60 

+ 2A.^ + &W-a 2 (2i2 + /2 2 ) =0. (31) 

161. Constant Susceptance Required for a Given Regulation. 
— For convenience in writing, let 

2 R + # 2 = m, m + 1 = (1 + #) 2 . (32) 

Then, by solving (31) for b , and combining terms, 



60 = ^ I A XB±\J(A x5) 2 -a 2 6 2 + 2^^ + ^o 2 |. (33) 

In equation (33) the constant term within the radical is 
(A xB)*-a?b 2 = -(A -B) 2 

and therefore, equation (33) reduces to 
bo = ^AxB±>J±(b>g + A-By-(l+^)(A.B)^ (34) 

162. Real and Imaginary Solutions. — In order that 6 be 
real, the quantity within the radical must be positive, 

i(6Vo + A. J B) 2 = (l + ^)(A. J B) 2 , (35) 

{Vg a + A-By 
+ m — (A • BY — ' (36) 

from which, by (32), 

(l + fl) 2 s(l + J^|) 2 - (37) 



CASE II 251 



Thus, for a real 6 with a positive R, 

b 2 

and for a real 6 with a negative R, 



b 2 
== # s j^g go, (a) 



-2 = R=-2-j^g . (b) 



(38) 



No real value of b can be found for a value of R which does not 
he within the above limits. In other words, for a given value of 
<7o, values of R not included within the above limits cannot be 
produced by shunting g with a constant susceptance, b . It is 
obvious from equation (33) that, for a finite g and R = 0, b 
becomes infinite. Under no condition, then, can zero voltage 
regulation be secured by such means, though as low a regulation as 
desired may be secured by the use of a condenser (or inductance, 
depending on the algebraic sign of bo taken from curves similar to 
Fig. 76) of sufficient size. These facts are emphasized, because 
in a recent engineering publication a method, based on incorrect 
ideas, was given for the determination of the constant value of 
JV to give zero regulation. Constant P } with zero voltage regu- 
lation implies a constant b , and according to the above, such a 
solution is impossible. 

163. Numerical Illustration. — Before further development 
from equation (34), let us see the form of curves resulting there- 
from, when applied to the numerical example just considered. 
Table XXXIV, below, gives values of 6 computed from (34), 
which, for the line constants used, becomes 



KF&o = 2.1141± y i(10 3 sr + 1.06610) 2 - 1.13657(l +~). (39) 
Both values of b resulting therefrom are tabulated. 



•_\-)2 



VOLTAGE REGULATION 



TABLE XXXIV 

Values of b = Load Susceptance Required to give a Regulation R with 
a Load Conductance, g , when g = at No Load and 6 Remains Con- 
• STANT. R as Defined by Equation (1). b X 10 s Tabulated 





0002 


0.0004 


0.0006 


0.0008 


0.0010 


0.0012 


0.0014 


0.0016 


0.0018 


0.0020 


0* 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


Inf. 


0.04 


-0.0259 
4.2541 


-1.2439 
5.4721 


-2.2394 
6.4676 


-3.1405 
7.3687 


-3.9890 
8.2172 


-4.8044 
9.0326 


-5.5972 
9.8254 


-6.3737 
10.6019 


-7.1379 
11.3661 


-7.8936 
12.1218 


0.08 


0.8232 
3.4050 


-0.0602 
4.2884 


-0.8380 
5.0662 


-1.4860 
5.7142 


-2.0915 
6.3197 


-2.6707 
6.8989 


-3.2320 
7.4602 


-3.7809 
8.0091 


-4.3202 
8.5484 


-4.8522 
9.0804 


0.12 


1.2793 
2.9489 


0.4274 
3.8008 


-0.1897 
4.4179 


-0.7292 
4.9574 


-1.2288 
5.4570 


-1.7045 
5.9327 


-2.1639 
6.3921 


-2.6128 
6.8410 


-3.0510 
7.2792 


-3.4849 
7.7131 


0.-16 


1.6525 
2.5757 


0.7746 
3.4536 


0.2149 
4.0133 


-0.2631 
4.4913 


-0.7013 
4.9295 


-1.1160 
5.3442 


-1.5152 
5.7434 


-1.9036 
6.1318 


-2.2841 
6 5123 


-2.6584 
6.8866 


0.20 


Imag. 


1.0346 
3.1936 


0.5050 
3.7232 


0.0660 
4.1622 


-0.3317 
4.5599 


-0.7057 
4.9339 


-1 0643 
5.2925 


-1.4122 
5.6404 


-1.7525 
5.9807 


-2.0867 
6.3149 


0.24 


I mag. 


1.2495 
2.9787 


0.7311 
3.4971 


0.3178 
3.9104 


-0.0514 
4.2796 


-0.3962 
4.6244 


-0.7252 
4.9534 


-1.0436 
5.2718 


-1.3541 
5.5823 


-1.6590 
5.8872 


0.28 


Imag. 


1.4433 
2.7849 


0.9177 
3.3105 


0.5210 
3.7072 


0.1725 
4.0557 


-0.1502 
4.3784 


-0.4567 
4.6849 


-0.7524 
4.9806 


-1.0403 
5.2685 


-1.3224 
5.5506 



164. Nature of Curves. — Plotted, this data gives the family 
of curves shown in Fig. 76. The curves are hyperbolas which 
become narrower as R increases. The true shape of the curves is 
not apparent, since the g and b scales are necessarily taken un- 
equal in order to make the curves readable. From equation (39), 
it is obvious that the displacement of the axis of the hyperbolas 
from the g axis is 2.114 X 10~ 3 . While in the preceding case (g 
and 6 disappearing simultaneously) the value b = 2.1141 X 10 -3 
gives a minimum regulation as pointed out, in this case, where & 
does not vanish at no load, an inspection of the curves shows that 
6 = 2.1141 X 10 -3 is the value which gives a maximum voltage 
regulation, with a given value of g . The left-hand branches of 
these hyperbolas are not shown, since a negative value of g is 
meaningless. 

Rationalizing equation (34), 

(0o + ^) ! -(2ft + R*)(6o-^J=(l + fi)'(^) ! - (40) 



CASE II 



253 



This is the equation of an hyperbola whose center is at the point 



A>B 



+ 



AxB 



(41) 



6 2 '62 

and whose axis is the horizontal line through this point (g being 




Fig. 76. — Relation between g and 6 for Any Value of R when 6 does not 
Vanish but Remains Constant at No Load. Equation (39). 

plotted horizontally). The vertices of the different curves then 
lie on this line, and their g coordinates are 



R 



A-B 



(42) 



for the different values of R. This value may be obtained from 
equation (38) by using the equality sign, which thus gives the 
minimum value of g for which a regulation R may be obtained. 



254 VOLTAGE REGULATION 

The asymptotes to the curves pass through the center (equation 

41), with a slope , =, and their equation is, therefore, 

v 2 R -{• R? 

. Ax B 1 / ,A-B\ .... 

The substitution of negative values of R according to equation 
(38) in equation (39) will yield the same family of curves as shown 
in Fig. 76. The curve for R = —2.16, for example, is the same 
as that for R = +0.16, but such solutions are meaningless for a 
negative regulation numerically greater than 1 has no significance. 

165. Significance of the Coordinates of the Central Point. — 
In this problem, the center for the system of hyperbolas coincides 
with the center for the system of circles in the foregoing case, and 
attention is again called to its location in terms of the short-circuit 
impedance of the line. The general equation on which these dis- 
cussions are based is 

E = AE + BI , and if we make E — 0, we have 

h A { ** } 

By E is meant the resulting terminal e.m.f. at the load end, so 
that if we consider an e.m.f. applied at the load end while the 
generator end is short circuited (E = 0), we have 

B _ applied voltage at load end 
A current produced at load end 

= short-circuit impedance measured at load end. (45) 

The reciprocal of this quantity is 

A _ di -f- j(h _ a>ibi + (hbi _ . aJh — <hbi _A'B .A X B 
B~ &i+j&2~ P J V~ ~P~~ 3 ~tf~ 

Iy 

= \-y coth VI = short-circuit admittance. (46) 

166. Mechanical Construction of Hyperbolas. — The two 
components of this short-circuit admittance are of opposite alge- 
braic sign to, but the same numerically as the two coordinates of 
the center of the curves in Figs. 74 and 76. Therefore, if the 
complex expression for this short-circuit admittance taken nega- 



CASE II 



255 



A pY 

tively, — -jt = —\-y coth VI, be laid off as a line on the curve 

sheets on which g and 6 are to be plotted in the same manner as 
a vector is drawn in a vector diagram, the center of the curves is at 



60 

\C/ B 






f \ 





\ \ g ° 



Fig. 77. — Graphical Construction of Curves Shown in Fig. 76. 



the end of the line. The circles for the first case may then be 
drawn according to equation (25), while the hyperbolas which 
give the solution for this second case may be drawn easily by a 
graphical process. 



256 



VOLTAGE REGULATION 



Through the curve center, located as above, or by equation (41), 
draw the straight lines which are the asymptotes of the hyperbola 
according to equation (43) for any particular value of R. In 
drawing these asymptotes, it is only necessary to locate one point 
on each line in addition to the center, C, and this is very easily 

done by using the known line slope, ± , Account 

v 2 R + R 2 
must be taken, of course, of the difference in the go and b scales, 





ye 










































R 


















> 






\\\ 
















O.L'l 




















l 


V 


\\\ 






























III 




























0.20 














/// 










\\\ 
































//'' 






\i 




















o.u; 
















/ / 








































// 




























0.12 














1 


/ 








































j 


/ 


























0.08 














/ 


1 1 




































































MM 









































































































v 


10 




















- 


'; 


- 


i 


- 


2 


l 




i 




: 


I 




1 




l 


) 


12 











































Fig. 78. — Relation between 6o and R for Different Values of 0o Cross-plotted 
from Fig. 76. Reading from Axis of Curves Outward, for go = 0.2, 0.4, 0.6, 
0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 X 10" 3 mhos. 

if such a difference exists. By equation (42), locate also the 
vertex of the hyperbola; 

Fig. 77 shows the graphical construction of these curves, which, 
for convenience in reading data, should be drawn immediately 
upon cross section paper. O is the origin of coordinates, deter- 
mined by the intersection of the g and 6 axes. C is the curve 

center, at the end of the line representing the quantity — r 5 , the 

short-circuit admittance taken negatively. The point A being at 
the vertex of the hyperbola, the distance BA is R times the dis- 



case in 257 

tance CB, according to equation (42). After having drawn the 
asymptotes and located the point A for any value of R in ques- 
tion, draw any line, QS, through the vertex A, and lay off the 
distance SP equal to AQ. The point P is then a point on the 
desired hyperbola. Continue in this way until a sufficient num- 
ber of points are obtained through which to draw the desired 
curve. This graphical construction remains valid even though the 
g and 60 scales are unequal, provided the points C and A are prop- 
erly located, and the asymptotes drawn according to equation (43). 

The family of curves in Fig. 76 could have been drawn by the 
above method instead of from the numerical data derived by com- 
putation from equation (39) and given in Table XXXIV. If the 
accuracy of plotted curves is sufficient, results may be obtained 
very rapidly by the use of this graphical process. 

By replotting from Fig. 76, the curves shown in Fig. 78 are 
obtained. 



Case EI. A Portion Only of the Load-end Susceptance 
Constant 

167. Load-end Voltages. — This case is a continuation in 

more general form of Case II. Here, the total load end admittance 

is divided into two parts — the load proper consisting of g and W, 

b ' 
which make up a load whose power-factor angle is <£o = tan -1 —, 

do 

and a constant load-end susceptance, &o, which does not disappear 
simultaneously with the load proper, and which represents ap- 
proximately a synchronous condenser. The voltage regulation 
under these conditions is determined by keeping the generator 
voltage, e, constant in magnitude, as well as b 0} and noting the 
variation in eo when g and &</ vanish. 

Following the same method as in Case II: 
Under load, Y = go + j (bo + W), 

Eo = a, + jo, + (go + jbo + jW) (bx + #5 ' ^ 7) 

Under no-load conditions, Y ' = + jb , the load-end voltage is 

Eo ' = a 1 +ja 2 -{-(0-{-jbo)(b 1 +jb 2 ) ' (48) 



258 VOLTAGE REGULATION 

If R = voltage regulation, 

e 2 
eo" = (1 + R) 2 eo' = fa-bM+fa + bJW 

(49) 



(a x + 6,0o - kW - k&o) 2 + («2 + 6200 + biV + V>o) 2 ' 

Simplifying according to the method used with equations (30), 
(31), and (32), where 

m = 2 R + R 2 , 

-mW + 2 [&V + m (aib 2 - a*f)i)] 60 + &W 

+ 2 (a^x + 0262) 0o + & V 2 - 2 (ai& 2 - 0261) 60' - ma 2 = 0, 
giving 

-m&W + 2 [b*W + mAxB]b + b 2 g 2 

+ 2A-Bg + WW 2 -2AX BW -ma 2 = 0. (50) 

168. Constant Portion of Load-end Susceptance for Given 
Regulation. — Solving the above for b Q , the constant amount of 
load-end susceptance required to give a regulation R with a load 
of conductance g and susceptance 60', 

. W . A x B 

bo — u — 

m b 2 



^V (1 + m) W 2 + mg 2 + 2m^ O - m 2 ^^ 
1(6V 



± 



V/^ W 2 + i (b 2 g + A . fl)» - (l + 1) (A • *)' J • (51) 

In equation (51), 6 ' is the susceptance of the load proper, and 
when in this expression b ' is zero, the solution for b , the suscept- 
ance inserted for purposes of voltage control, corresponds with the 
solution given in equation (34) for 6 under such a special condition. 

As it stands, equation (51) involves two independent variables, 
0o and W, which specify the load conditions, and therefore it is 
impracticable to plot a sufficient number of curves to cover all of 
the possibilities therein. 

If, in any particular case, the value of W is known for every value 
of 00 (as, for example, with a load of constant power-factor, where 
W = 00 tan <fo), the labor involved in computing a set of data from 



CASE III 259 

which to plot curves similar to those in Fig. 76 is not excessive. 
The condition of constant power-factor of load proper will be taken 
up later. 

169. Nature and Location of Curves. — Equation (51) assumes 
the form 

which is the equation of an hyperbola in the variables g and b 
when &</ is constant. The coordinates of the center of the hyper- 
bola are 

A ' B * n A V , A XB 

"JT and 2R + & + -V~-' (53) 

which are the same as given in equation (41) for a non-inductive 
load, with the exception of the amount „ °. D „ , which, in this 

Z K t ti 

case, is added to the & coordinate of the central point. Changing 
the value of 6 ' thus causes the centers of the curves to take dif- 
ferent positions along the perpendicular line passing through the 

end of the vector — ^ . The displacement of the center from this 

D 

point is 

2 R + # 2 ' ^ 

The distance from the center to the vertex of the hyperbola is 



aW(nr) 2 -2TO' < 55 > 

and, as in the preceding case, the asymptotes to the curve have 
slopes 

± , * (56) 

V2R + R 2 

170. Mechanical Construction of the Curves. — Equations 
(54), (55), and (56) enable the hyperbolas to be plotted by the 
convenient graphical construction for any fixed values of R and 
W, though in this case the center of the curve changes its position 
with each change in the above quantities, according to equations 
(53) and (54). 



200 



VOLTAGE REGULATION 



The value of the radical in equation (55) may conveniently be 
found from the simple right-angled triangle relation. Draw two 
lines intersecting at right angles, and from their intersection lay off 



W 



along one of them, and with a radius 



A-B 



describe 



V2R + K 2 

an arc intersecting the other. The intercept is then the desired 

value of the radical. 

Fig. 79 shows the relations existing in the diagram giving b as a 
function of go, for fixed values of R and &</• Having the asymptotes 



(^^^M 




Fig. 79. — Graphical Construction for Plotting the Hyperbolas Represented 
by Equations (51) and (52). 



and the vertex of the hyperbola as obtained by the construction 
indicated in this figure, the curve may be completed by the me- 
chanical process previously described, Fig. 77. 

In any given case, the most convenient procedure probably 
would be to plot on one curve sheet a number of such curves for 
different values of b ' = susceptance of load proper, but all for a 
fixed value of regulation R. For all of the curves, the asymptotes 
would then have equal slopes — the center and vertex only chang- 
ing with changes in b '. By cross-plotting from a number of such 



CASE III 



261 



curve sheets, each of which is constructed for a different value of 
R, almost any desired information may be obtained. Fig. 80 
shows such a set of curves applying to the same transmission line 
as treated by Fig. 76, for the particular case, R = 0.16. The 
hyperbolas for &</ = 0.0006, 0.0004, 0.000, -0.0004 and -0.0006 
are drawn. The curve for & ' = 0.000 is the same as that shown in 
Fig. 76 for R = 0.16, the difference in their appearance arising 
from a difference in the scales used in the two curves. 



*>o 


xl( 


' 




1 






/ 


/ 


' ■ 


























/ 




/ 


















4.S 








A 


/ 


/ 




























/ 




/ 




















4.0 








/ 




f 


















— 









f- 


-" 


-7 










— 


— 






— 




































u 
































8,4 










\\ 


V 




















































_ 




l.fi 










/i 


\ 






























'/ 




















— 














i 










_. 






, 









































i 




i 










O 3 


10 


! 










-1 


(i 


h 


sV 









).S ' 


Ai 


B 


2 


i 


8 


2 




-0.8 
































































-1.0 












\ 
































\ 


































\ 

















Axes for 

0.6 x 10 s 
0.4 x 10 s 



■0.4 X 10 s 
0.6. r^lO 8 



Fig. 80. — Relation between g and b for R = 0.16 and for Different Values 
of W. Locate the Different Curves by Their Axes, which are Numbered 
with the Corresponding Values of &</• 

171. Forms of Curves in Special Cases. '— As is to be noted 
from Fig. 80, the hyperbolas have greater curvatures at their 
vertices for the larger numerical values of &</• By equation (55), 
when 

&o /2 =(2# + fi 2 )(^J, (57) 



262 VOLTAGE REGULATION 

the distance from the center to the vertex of the hyperbola is zero, 
and therefore the curve reduces to two intersecting straight lines — 
the asymptotes. For this particular case, the bo coordinate of 
the center of the curve is 

^±^=^£ (58) 

from equation (53), and according to whether W is condensive or 
inductive the algebraic sign in (58) is positive or negative. 

For values of 6 ' greater than that indicated by equation (57) 
the right-hand member of equation (52) becomes negative, and 
the equation thereby represents hyperbolas whose major axes 
coincide with the line passing perpendicularly through the end of 

the vector — ■= . The centers of the hyperbolas are located as 

before, according to equation (53), but the vertices, now falling 
upon the perpendicular line through the center, are at distances 



1 + R 
2R + R* 



\Jw*-(2R + R*)(^>j (59) 



from the central points. The right-hand sides of both branches of 
these hyperbolas must be drawn in order to furnish complete in- 
formation by means of the double value of b secured for each value 
of 0o, while in the case of the hyperbolas whose major axes are 
horizontal, the double value of 6 is secured from both sides of the 
right-hand branches. 

In the particular numerical problem previously used for illus- 

A 

tration, where - -g = - 0.0010661 +J 0.0021141, by equation 

(57) the hyperbolas reduce to straight lines for 

W = ± 0.0010661 V2 R + R?. 

Taking R = 0.16 as in Fig. 80, W = ± 0.000627 mho for such 
a condition. In Fig. 81 the curves for W = 0.000, 0.0004, 
0.000627, and 0.0010 are shown. 

172. Zero Voltage Regulation. — In this case — constant 
susceptance of load proper — a condition of zero voltage regula- 
tion can be secured as long as &</ has a finite value. The graphical 
process of plotting the curves cannot be applied for 72 = because 
the centers of the curves fall at an infinite distance above the go 



CASE III 



263 



axis and the asymptotes have slopes of ± infinity. Introducing 
into equation (50) the condition that m = 2 R + R? = 0, and 
solving for &</, 



.1 = 

_|ft=0 



AXB W A • B g 



fr 2 W 2 6 



(60) 







\ 


bx 


io :! 




























Jl2 




















) 










































10 
































































8 J 
































































6 














A 


xes 


for 


ft' 












A 




/— 


— 





— 





— 







K310C 


I 









0.00062 
























"qooo4o[i* 







































It 


























A 
















(/o ' 


io 3 












f 




i 1 


\ 




2 




4 









3 




10 


12 






S" 2 






























_j 
































i 


-4 


























































■ 






-0 






















1 





































Fig. 81. — Relation between g and b for R = 0.16 and Different Values 
of 6 '. The Curves Become Straight Lines for 6 ' = 0.000627. 



the equation of a parabola in g and 6o- Numerical data for curve 
plotting is so easily obtained from the above equation that graph- 
ical methods of construction need not be developed. For every 
real value of g as well as for every real value of &</ a real solution 
for 6 is obtained. No imaginary solutions, indicating impossible 
physical conditions, are to be obtained. 

In general, when negative values] of R are considered, corre- 
sponding to a decrease in load-end voltage when the load proper 



264 VOLTAGE REGULATION 

is removed, it is easily seen by equation (52) that the curve re- 
lating 0o and &o is an ellipse — throughout certain ranges in the 
value of R. Being of lesser ^importance, detailed development of 
this matter is not given here. 

Case IV. The Load Proper of Constant Power-Factor 

173. Equation Relating Constant Susceptance and Regula- 
tion. — Under this head is considered a load proper the two 
components of whose admittance are always in a constant ratio to 
each other. The power-factor angle of the load is then constant, 
and the problem is to determine the proper constant amount of 
additional susceptance at the load end to produce certain values 
of voltage regulation. 

Let 



0o = power-factor angle of load proper, counted positive 
for a leading load. 

k = tan 0o = — , W = kg . 
0o 



(61) 



go and b r are the two components of the load admittance. The 
above value of W may be substituted directly in equation (50), 
giving 



(1 + V) gfo 2 + 2 kgobo - m6o 2 + 2 (^ - k ^jffi 



ft 



+ 2m^J*-«^0. (62) 

174. Location of Curves. — (62) is a general equation of the 
second degree in g and & as variables, and it therefore represents 
as a curve some conic section whose center is displaced from the 
origin of coordinates and whose axis is angularly displaced from 
the coordinate axes. 

From the coefficients of this equation and by the methods of 
analytical geometry we find that the major axis of the conic 
section or curve makes an angle, a, with the g axis (counted 
counter clockwise) determined by 

tan 2 « = 1+ l*+A.' a = \^- l JT+W+¥' (63) 



CASE IV 



265 



The center of the curve is displaced from the origin of coordinates 
by the amounts 

2 R + R 2 A-B 



and 



(1 + R) 2 k 2 + 2R + R 2 
AxB k 



6 2 



A>B 

b 2 



(64) 



b 2 (l + R) 2 k 2 + 2R + R 2 

along the g and 6 axes, respectively. 

175. Nature of Curves. — The nature of the curve represent 
ing equation (62) is determined as follows: 

< 0, an ellipse, 
(1 + R) 2 k 2 + 2 R + R 2 



= 0, a parabola, 
> 0, an hyperbola. 



(65) 



Since negative values of R are seldom considered and since k 
enters into these determining conditions only as k 2 , the left- 
hand member in the expression (65) is usually positive and greater 
than zero. The curve is then usually an hyperbola, and graphical 
methods of construction of the curve for this case only will be 
considered. To determine the slopes of the asymptotes to the 
hyperbola it is only necessary to determine the slopes of the curve 
represented by equation (62) at infinity. Since g and 6 have 
infinite values at an infinite distance from the origin, the first 
powers of these variables as well as the constant term in the 
equation (62) may be neglected in comparison with their squares 
and product. 

Thus, as approximation at infinity, 

(l+fc 2 )0o 2 +2fc<7o&o-m&o 2 = O, <7o 2 + 7^5 9oh - r^T5 &o 2 = 0. (66) 



1 + k 2 



1 + k 2 



Assuming this equation to consist of the product of two linear 
factors, as 

(go + X&o) (00 + yb Q ) = 0, 1 

go 2 + (X + y) gob + X7&0 2 = 0, J K } 

and equating coefficients of like terms in equations (66) and (67), 



X = 



7 = 



1c + Vk 2 + m (1 + k 2 ) 
l + k 2 
Vk 2 + m (1 + A; 2 ) 
1 + & 2 



(68) 



266 VOLTAGE REGULATION 

From the two equations, 

go + X&o = and g + 7&o = 0, 

and the relation, m = 2 R + R 2 , the slopes of the asymptotes are 
determined as 

l + fc» 



Slopes of asymptotes = — 

k d= "v kt -f m ^i -j- «*; 
i _i_ 1.2 

(69) 



A; ± VK* + m (1 + k 2 ) 

1 + fc 2 



fc ± V ( 1 + R) 2 k 2 + 2 # + it! 2 

176. Mechanical Construction. — Knowing, from equation 
(64), the position of the center of the curve and from equation 
(69) the slopes of the asymptotes, the hyperbola is completely 
determined if the position of one point on the curve is known. 
Two such points may be determined by placing 6 = in equation 
(62) and solving for g . 

Thus, for 6 = 0, 

±V(A -B-kAxB) 2 + m{\ + k 2 ) a 2 b 2 \. (70) 

From the data in equations (63), (64), (69), and (70), the hyper- 
bola may be constructed by the graphical or mechanical process 
described in Fig. 77, and the labor involved is not excessive. 

In this connection it is well to note that the axis and asymptotes 
should be drawn first, for then it may be possible to determine by 
inspection whether the two points on the curve, as located by 
equation (70), will be suitably situated for an accurate continuation 
of the mechanical process of curve construction. If the two 
points so located should happen to fall very near to the asymp- 
totes (as compared with the distances from the points to the 
center) accurate construction by the previously described method 
is not readily obtained. In such cases, a value of 6 equal approxi- 
mately to the ordinate of the center of the curves may be substi- 
tuted in equation (62) and the corresponding values of go deter- 
mined by solving the resulting quadratic equation. These points 
will generally fall near the vertices of the curves — the ideal 
condition as regards the accuracy of the graphical process of curve 
drawing. 



CASE IV 



267 



177. Numerical Illustration. — Figure 82 shows the hyperbola 
giving the relation between g and b Q for the numerical case under 
discussion, for the arbitrarily selected conditions: R = 0.16 and 
k = tan <£o = —0.75, which corresponds to a lagging load of 0.80 















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4 






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I 






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\ 


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V 

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: 10 


i 




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Fig. 82. — Load-end Snsceptance b Required for a Regulation of 0.16 
at a Constant Power-Factor of Load of 0.80, Lagging. 

power-factor. The data from which the curves may be constructed 
are: 

Angle, a, between axis of curves and go axis = —19.086 deg. 
Angles between asymptotes of curves and g axis, +40.959 deg. 

and -79.131 deg. 
Coordinates of center, -0.0003342 and +0.002839. 
0o for 6 = 0, -0.0037268 and +0.0003327. 
g for 6 = 0.00300, -0.0009578 and +0.0004437. 

In this particular case the positions of the points as determined 
for 6o = were not suitable for the process of graphical con- 



268 'VOLTAGE REGULATION 

struction. The positions of the two points for 6 = 0.00300 were 
therefore computed, and the points so located used as a basis for 
further construction. 

The foregoing methods for the determination and construction 
of voltage regulation curves for transmission systems are merely 
illustrative of general methods which may be employed. In 
regard to the determination of the proper load-end susceptance 
to give specified values of regulation, a number of special condi- 
tions have been considered, but it is obviously impossible to take 
up individually the great number of such conditions which can 
arise. In a subsequent volume, dealing primarily with power 
transmission, a more complete discussion is contemplated. It is 
obvious that similar developments covering the transmission of 
power by means of constant, or nearly constant, current are 
possible. 



APPENDIX 

HYPERBOLIC FUNCTIONS 

A description of the hyperbolic functions with their geometrical 
significance is not required in order to lead to a clear understanding 
of their use in the analytical expressions of the solutions of physical 
problems. For such purposes, the definitions 

. e x + e~ x .x 2 x* 

. coshx=— 2— = 1+2+24+ ••' , 



pX __ p — X yjr»3 /v»5 

sinhz =—£- - = ^+6 + 120 + 

, sinhz x 3 . 2x 5 

tanh x = — r— = x — «" + te" r 

coshz 3 15 



(1) 



usually suffice. Familiarity with the types of differential equa- 
tions whose solutions are expressible in terms of these functions 
is desirable. Fig. 83 shows the curves representing the hyper- 
bolic functions of a real variable, and as is apparent from an 
inspection of the series for these functions, their nature is essentially 
more simple than that of the trigonometric functions, since they 
are not periodic. 

An excellent working table o the natural values and logarithms 
of these functions is given by Becker and Van Orstrand in the 
"Smithsonian Mathematical Tables." The values are given to 
five decimal places. A collection of formulae relating to the 
hyperbolic functions is also given by them, and on account of its 
usefulness it is, by the permission of the publishers, reproduced 
here in part. A very useful table of the hyperbolic functions of 
complex variables has recently been published by Professor A. E. 
Kennelly. It is particularly useful in the rapid determination of 
approximate solutions, but for very precise work, the double 
interpolations involved render its application cumbersome. 

As before stated, a five-place accuracy in the tabulated values 
of functions is sufficient for the majority of physical and engineer- 
ing calculations, but since a table of hyperbolic functions is not 

269 



270 



APPENDIX 



used as frequently as tables of trigonometric functions or common 
logarithms, it appeared desirable to the writer to prepare and 
publish herewith a six-place table of the logarithms of hyperbolic 
functions. They provide for the exceptional case which requires 
the degree of precision afforded by logarithmic computations to 

















































































2.0 
































].N 






















\c 


iosii 


X 






l.fi 
































1.4 
































u 
































1.0 












Sin 


l X 


















.8 












Ta 


ih^r 


















.fi 
































.1 
































.2 



















































-1 


;-i 


2-1 


o -. 


B - 


6 - 


i -. 


1 / 




> . 


1 . 


r, 


>s 1 


1 


2 I 


4 
















-A 
































-.0 




















Ta 


ill 1 










-s 
































-1.0 






















' 










-1.2 
































-l.f 






















Sin' 


1 X 








-l.G 



















Fig. 83. — The Hyperbolic Functions of a Real Variable. 



six significant figures, and at the same time suffice, without an 
undue increase in the labor of interpolation, for five-place com- 
putations. 

The six-place tables which follow were taken from twelve- 
place tables which were computed by B. M. Woods and the 
writer, and which appear in the Engineering Series of Publications 
of the University of California. Tabulations of logio cosh x, 

logio sinh x, logio tanh x, logio , and logio 1 — r— are given for 



APPENDIX 



271 



values of x from 0.000 to 2.000, which range is ample for the 

greater portion of engineering problems of the nature treated in 

this work. Tabular intervals of 0.001 in the argument and the 

x 

render interpolation 



sinh x 
use of the quantities log and log 



tanhx 



easy in all portions of the table. 

SI 111 1 jC 

The quantities log - ; — and log 



are given for values of 



x ~ tanh x 

x from 0.000 to 0.500, and serve for the determination of log sinh x 
and log tanh x throughout this range where otherwise the inter- 
polations would be troublesome. Thus, given log x and x: 

sinhx 



and 



log sinh x = log x + log 



log tanh x = log x — log 



tanhx 



(2) 



Given log sinh x or log tanh x, interpolations by proportional 

x 



sinh x 
parts may be made for log — - — and log 



x " tanh x 

differences for the latter quantities are small. Thus, 

sinhz 



, since the tabular 



and 



log x = log sinh x — log : 



log x = log tanh x + log 



tanhx 



(3) 



The method of use of the auxiliary tables is identical with that 
of the "S and T" tables used for the determination of the loga- 
rithms of the trigonometric sines and tangents of small angles. 

In the portion of the table from x = 0.500 to x = 2.000, the 
differences between successive values are tabulated, and it is to 
be noted that these differences are such that linear interpolations 
only are required. 



272 APPENDIX 

Relationships Involving Hyperbolic and Allied Functions 

Reproduced by permission from the Smithsonian Mathematical Tables 

A. — Relations between Hyperbolic and Circular 
Functions 

1. sinh u = —i sin iu = tan gd u. 

2. cosh u = cos iu = sec gd u. 

3. tanh u = — i tan iu = sin gd u. 

4. tanh \ u = tan \ gd u. 

5. e" = (1 + sin gd u) + cos gd u 

= [1 — cos (| 7r + gd u)] + sin (£ 7r + </d w) 
= tan (j 7r + I gd u). 

6. sinh iu = i sin u. 

7. cosh iu = cos u. 

8. tanh iu = i tan u. 

9. sinh (u ± £y) = ± i sin (y =F iu) 

= sinh it cos v ± i cosh m sin y. 

10. cosh (u ± iv) = cos (y Tiw) 

= cosh u cos y ± i sinh u sin y. 

11. cosh (miir) = cos m*-. (m is an integer.) 

12. sinh (2 m + 1) \ in = i sin (2 m + 1) | x. (w is an integer.) 

B. — Relations among the Hyperbolic Functions 

13. sinhu = \ (e u — e - ") = —sinh ( — u) = (cschu) -1 

= 2 tanh | m 4- (1— tanh 2 \u) = tanh u 4- (1— tanh 2 u)*. 

14. cosh u = \ (e u + e~") = cosh (— m) = (sech w) -1 

= (1 + tanh4u) 4- (l-tanh^u) = 1 4-(l-tanh 2 u)*. 

15. tanh u = (e u - e~ u ) 4- (e u + e - ") = —tanh (— u) 

= (coth m) _1 = sinh u 4- cosh w = (1 — sech 2 u)^. 

16. sech u = sech (— u) = (1 — tanh 2 u) 5 . 

17. cschu = — csch (— u) = (coth 2 u — 1)^. 

18. coth u = —coth (— u) = (csch 2 u -\- 1)K 

19. cosh 2 u — sinh 2 u = 1. 

20. sinh £ u = V^ (cosh u — 1). 

21. cosh |u= V^ (cosh w + 1). 

22. tanh \u = (cosh u — 1) 4- sinh u 

= sinh u-7- (1+cosh u) = \/(cosh w— 1) 4- (cosh w+1). 



APPENDIX 273 

23. sinh 2 u = 2 sinh u cosh u = 2 tanh u-f (1- tanh 2 w). 

24. cosh 2w = cosh 2 w + sinh 2 u = 2 cosh 2 w — 1 

= 1 + 2 sinh 2 u = (1 + tanh 2 u) 4- (1 - tanh 2 u). 

25. tanh 2 w = 2 tanh w + (1 + tanh 2 u). 

26. sinh 3 u = 3 sinh u + 4 sinh 3 u. 

27. cosh.3 w = 4 cosh 3 w — 3 cosh u. 

28. tanh 3 u = (3 tanh u + tanh 3 u) 4- (1 + 3 tanh 2 u). 

29. sinh mt = 

ii-i . (n) (n— 1) (n — 2) , , . , _ 

n cosh n_1 m sinh u+ —^ -pr-^ cosh" -3 u sinh 3 u -f- • • • . 

6 

30. cosh nu — cosh" u -\ *-= cosh 71-2 w sinh 2 it 4> • • • . 

31. sinh w + sinh v = 2 sinh | (it + y) cosh § (w — y). 

32. sinh it — sinh v = 2 cosh | (w + w) sinh ^ (m — y). 

33. cosh it + cosh v = 2 cosh ^ (w + y) cosh \ (u — v). 

34. cosh u — cosh v = 2 sinh | (it + y) sinh | (u — v). 

35. sinh u + cosh it = (1 + tanh J u) 4- (1 — tanh £ it). 

36. (sinh it + cosh u) n = cosh nw + sinh ww. 

37. tanh u + tanh y = sinh (u + y) 4- cosh it cosh y. 

38. tanh u — tanh y = sinh (it — v) 4- cosh w cosh y. 

39. coth u + coth y = sinh (it + y) + sinh it sinh y. 

40. coth it — coth y = — sinh (w — y) 4- sinh w sinh v, 

41. sinh (it ± y) = sinh u cosh y ± cosh u sinh y. 

42. cosh (u ± y) = cosh w cosh y ± sinh u sinh y. 

43. tanh (it ± y) = (tanh w ± tanh y) 4- (1 ± tanh w tanh y). 

44. coth (it ± y) = (coth w coth v ± 1) 4- (coth y ± coth it). 

45. sinh (it + y) + sinh (u — v) = 2 sinh it cosh y. 

46. sinh (it + y) — sinh (w — y) = 2 cosh u sinh y. 

47. cosh (u -f- y) + cosh (it — y) = 2 cosh u cosh y. 

48. cosh (it 4> y) — cosh (it — y) = 2 sinh it sinh y. 

49. tanh \ (it + y) = (sinh w + sinh y) 4- (cosh it + cosh y). 

50. tanh \ (it — v) = (sinh u — sinh y) 4- (cosh u + cosh y). 

51. coth I (w + v) = (sinh it — sinh y) 4- (cosh u — cosh y). 

52. coth \ (u — y) = (sinh u + sinh y) 4- (cosh m — cosh y). 
_„ tanh m + tanh y _ sinh (u + y) 

tanh m — tanh y sinh (w — v) 
K . coth u + coth y _ _ sinh (u + y) 

coth u — coth y sinh (u — y) 

55. sinh (u + y) + cosh (u + y) = (cosh u + sinh w) (cosh y + sinh y) . 



274 APPENDIX 

56. sinh (u -f v) sinh (u — v) = sinh 2 u — sinh 8 v 

= cosh 2 u — cosh 2 v. 

57. cosh (it -+- v) cosh (u — v) = cosh 2 u + sinh 2 v 

= sinh 2 u + cosh 2 v. 

58. sinh (miV) = 0. (m is an integer.) 

59. cosh {mix) = ( — l) m . 

60. tanh (miir) = 0. 

61. sinh (u -f- mzV) = (— l) m sinh u. 

62. cosh, (u -f- rraV) = ( — l) m cosh u. 

63. sinh (2m+l)} Mr = ±t\ 

64. cosh (2 m + 1) \ Mr = 0. 

65. sinh f y ± u J = t cosh u. 

66. cosh [ y ± w) = ±i sinh u. 

67. tanh (u + tV) = tanh u. 

C. — Inverse Hyperbolic Functions 

68. sinh- 1 u = log (if +Vu 2 + l) = cosh" 1 Vi?+1 = f — ^— , • 

^ (u 2 + l)* 

69. cosh- 1 u = log (u + Vu 2 - l) - sinh- 1 v^l = C—^L— . 

J (u 2 -l)* 

70. tanh" 1 u = \ log (1 + u) -\ log (1 - u) = fj^' 

71. coth- 1 M = ^log(l + W )-^log(M-l)= f-^- 2 = tanh" 1 -- 

J 1 — ir u 

72. sech- 1 M=logf-+v/-\-l) = - f ^— ^cosh- 1 ^ 

V« y w 2 / J u(l -u 2 )* u 

73. csch- 1 u=log(- + V^ + l) = - f — r = sinh" 1 -- 

74. sin -1 u = — isinh -1 iu = — i log (iu + Vl — u 2 ). 

75. cos -1 u = — i cosh -1 u = — i log (u + i Vl,— u 2 ) . 

76. tan -1 u = — i tanh -1 iu = 75-. log (1 + w) — —. log (1 — iu). 

77. cot -1 u = i coth -1 iu = =-? log (iu — 1) — 7r . log (iu + 1). 

78. sin -1 iu = i sinh -1 u = i log (u + Vl -f- u 2 ). 



APPENDIX 



275 



79. cos -1 iu = — i cosh -1 iu = g — i log (m + vl + w 2 ). 

i % 

80. tan -1 iu = i tanh -1 u = = log (1 + w) — ~ log (1 — w). 

81. cot -1 iu = —i coth -1 w = — 5 log (u + 1) + 5 log (w — 1). 

82. cosh" 1 h(u + i) = sinh" 1 § ( w - ^j = tanh- 1 grjOJ 

= 2 tanh -1 — r-= = log w. 

M + 1 

83. tanh -1 tan u — \gd2u. 

84. tan^ 1 tanh u = \ gd~ l 2 w. 

85. cosh -1 esc 2 m = — sinh -1 cot 2 w = — tanh -1 cos 2 w = log tan u. 

86. tanh -1 tan 2 ( j x + \ u) = \ log esc w. 

87. tanh -1 tan 2 % u = \ log sec it. 

88. cosh -1 u ± cosh -1 v = cosh -1 [uv ± V(u 2 — 1) (v 2 — 1)]. 

89. sinh" 1 u ± sinh" 1 = sinh" 1 [u Vl + y 2 =b i> vT+m 2 ]. 



D. — Series 

AA ^ 1 . 14/ . 14 . 14/ . 

90. e u = l+" + 2! + 3l + 4l+ '"• 

91. logu=(M-l)-|(u-l) 2 + i(M-l) 3 - • • • . 

»^^ + "lP=i)r + ip=i)r + ... 

94. log (1 + u) = u — g M 2 + ~ w 3 — 7 M 4 + • • • . 

95. log(j^)=2[ M + | W 3 + jU 5 + ^u 7 + • • -J. 

w 3 u^ u} 

97. sinh m = m + 31 + 5"j + t]+ ' * * 

= "( 1 ^)( 1+ p)( 1 + 3^)---; 

98. coshw=l+^ + ^ + |-j+ • .- 



(m 2 < 00 . 

(2>m>0. 

. (u > \. 

1. (m>0. 

(m 2 < 1. 

(u 2 < 1. 

(u 2 > 1. 

(u 2 < 00 . 

(u 2 < 00 . 

(u z < 00 . 



276 



APPENDIX 
"*" 3^7 V 1 + 5*^J 



hmhmhm 



99. tanh u = a- s u 5 + 



— «* - 



L5 



u- 



V7_ 
315 



u 7 + 



100. u coth u=l+„-u ! -^,u 4 +J;« 6 - 
3 45 94d 



101. sech u=l-n« ! + 



21 



u l 



61 
720 



u 6 + 
31 



102. M cschu=l-^u^3^^- 1 



u 6 + 



103. ^^ = <^ = ^-6" 3 + 24" 5 -5oTo w7 + 
x , 1 sech 3 m 13 sech 6 it 
= __ sech u-2-3 24-5-- 



104. i« = flK*- l * = * + |*» + ^ 



61 



5040 



<*> 7 + 



in _ . , . 1 u 3 , 1 3 m 5 1 3 5 m 7 , 

105. ^ 1 « = «- 2 3+245-2467 + -' 

1 1 13 1 135 1 
g + 22m 2 244u 4 + 2466u 6 

1M ._, . 11 13 1 135 1 

106. cosh ^=1082^-2^-244^-2466^- 

107. tanh- 1 ** = u + 5 u* + \ u* + \ u 7 + • • • . 

6 7 

108. coth- 1 u=tanh- 1 - = - + ^ 3 + ^- 5 + ; ^-:+ • . 

u u 3u 3 5 m 5 7w 7 

109. sech iu=cosh i- = l g-----244 -^ee " 



(u 2 < 00 .) 
(m 2 <1t«. 

(M 2 < X 2 . 
(W 2 <7T 2 . 

(m small.] 
. (u large. 

(u 2 < 10 
. (u 2 >l. 
•• (w 2 >l. 

(u 2 < 1. 
. (u 2 > 1. 



lia. csch -1 u=sinh- 1 - = ^5— -f ~t-=— - 6 — st a^t^f "+ 

u u 23u 3 245 w 6 2467 m 7 



(m 2 < 1. 



= 1 2, 1m 2 _13m| 135«;_ 
g u + 2 2 244 + 2466 



(w 2 > 1. 
. (w 2 <l.) 



E. — Derivatives 



111. ^- = e\ 

du 



112. d 



lo&iM 

du 



u 



da v dv , 

113 - iH =a -Tu- log ' a - 

114. ^ =««(! + log. «). 



APPENDIX 277 

, ., _ d sinh u , . ., „ d tanh u , „ 

115. — j = coshw. 117. 5 = sech 2 M. 

du du 

„.,- d cosh w . . ,„ rf coth u ,. 

116. ; = sinnw. 118. 5 = — csch 2 w. 

du du 

119. — -i = — sechw. tanhw. 

du 

120. — -, = — cschu. cothw. 

du 

101 d sinh -1 u _ 1 10K dsech _1 M 



<*U V« 2 + 1 du u y/l 

^ nn d cosh -1 u 1 , M d csch -1 w 

122. ; = ■ ; 12b. 



du Vu 2 - 1 ^w w Vu 2 + 1 

nh" 1 m = 1 127 rfgdi 
du 1 — u 2 du 

>th -1 u = 1 12g rfgd- 
dw 1 — w 2 ' du 



00 dtanh 1 !* 1 10 _ dgdw , 

123. j = : =• 127. —3 — = sechw. 

du 1 — u 2 du 

10 . dcoth _1 M 1 100 dgd _1 w 

124. ; = ; 5- 128. -~ = sec u. 



F. — Integrals. (Integration constants are 

OMITTED.) 

129. I sinh udu = cosh u. 131. I tanh udu = log cosh u. 

130. / cosh u du = sinh w. 132. I coth udu = log sinh u. 
133. / sech udu = 2 tan -1 e" = gd u. 

135:. / csch udu = log tanh 5 • 

135. / sinh n u du = - sinh n_1 w. cosh u / sinh 71-2 u du 

J n n» J 

= — r-s sinh n+1 u cosh u f-s / sinh n+2 u du. 

n + 1 w+ 1 J 

136. / cosh n u dw = - sinh u. cosh n_1 u -\ / cosh" -2 u du 

J n n J 

1 n + 2 C 

= r-r sinh u cosh n+1 u H :— 7 I cosh n+2 u du. 

, n+ 1 n+ 1 J 

137. / m sinh udu = u cosh w — sinh u. 

138. / m cosh udu = u sinh w — cosh it. 



278 APPENDIX 

39. I m 2 sinh udu = (u 2 + 2) cosh u — 2 u sinh u. 

40. / u n sinh udu = u n cosh u — nu n ~ l sinh w 

+ n (n — 1) / u n- a sinh u dw. 

41. / sinh 2 u du = \ (sinh u cosh u — u). 

42. I sinh u. cosh u du = \ cosh (2 u). 

43. / cosh 2 u du = \ (sinh u cosh u + u). 

44. I tanh 2 udu = u — tanh u. 
'45. I coth 2 udu = u — coth u. 

46. / sech 2 udu = tanh u. 

47. / sech 3 udu = \ sech u tanh u -}- $ gd u. 
. j csch 2 udu = — coth u. 

I sinh -1 udu = u sinh -1 u — (1 + u 2 )*. 
. / cosh -1 udu = u cosh -1 u — (u 2 — 1)*. 

51. I tanh -1 udu = u tanh -1 u + \ log (1 — u 2 ). 

52. J u sinh -1 u du - | [(2 u 2 + 1) sinh -1 u - u (1 + u 2 )*]. 

53. fu cosh -1 u du - I [(2 u 2 - 1) cosh -1 u-u(u 2 - 1)*]. 

54. / (cosh a+ cosh u) -1 du = 2 csch a. tanh -1 (tanh £ u. tanh \ a) 

= csch a [log cosh | (w + o) - log cosh \{u — a)]. 

55. / (cos a -f cosh u) -1 du = 2 esc a. tan -1 (tanh £ u. tan \ a). 

56. / (1 + cos a. cosh u) -1 du = 2 esc a. tanh -1 (tanh \ u. tan \ a). 



APPENDIX 279 

157. / sinh u cos u du = \ (cosh u. cos u + sinh u. sin u). 

158. / cosh u. cos udu = \ (sinh u. cos w + cosh u. sin u). 

159. / sinh u. sin udu = \ (cosh w. sin u — sinh w. cos u). 

160. / cosh m. sin u du = | (sinh w. sin w — cosh u. cos u). 

161. / sinh (mu) sinh (nw) du 
-j— — ^ [m sinh (nw) cosh (mw) — n cosh (nit) sinh (mw)]. 



mf — n" 



162. I cosh {mu) sinh (nw) dw 



— — — - 2 [w sinh (nw) sinh (mu) — n cosh (nw) cosh (mu)]. 



163. / cosh (mu) cosh (nu) du 



2 — — 2 [m sinh (mu) cosh (nw) — n sinh (nit) cosh (raw) 1 . 



wr — rr 



164. / sinh u tanh udu = sinh u — gdu. 

165. I cosh m coth udu = cosh w + log tanh ^ • 

166. I sec it dw = gd _1 u. 

167. lsec 3 0d</>= / (1 + tan 2 0)^dtan<£ = ^sec0tan<^ + |gd~ '</> 

= \ tan 0(1 + tan 2 <t>) a + ? sinh -1 (tan <£). Here tf> = gd u. 

ico C du . . ,u f* du . ,u 

168. I r = sinh * -• I r = sin -1 -• 

•/ (u 2 + a 2 )* a J (a 2 -u 2 )? a 

189. r_^L_- = cosh-^. r_^^_=co S -^. 

J (u 2 - a 2 ) 5 a J (a 2 - w 2 ) 5 a 

170. f ,, d \ -I**** f^,= lta^. 

J (a 2 — w 2 ) u<a a a J a? -{- u 2 a a 

J (ir — a 2 ) u>a a a J a 



—du 1 , i w 
- = -cot -1 — 



2 + m 2 a a 



172 . f ~ d » , -iwelr*". f ^—.-iwc-tH. 



280 APPENDIX 



173. f - dU . -leech-'?. f ~ d " , -JMB-a. 

»/u(a 2 + u 2 )' a a J M (u 2 -a 2 )* a o 

ii a C du 1 . , , au + i , 

174. I r = — ■ t=. sinh _1 j- (a positive, ac> ft 2 .) 

J (au 2 +2 6u+c)* v^ (ac-ft 2 )* ^ »w<^«r.j 

1 ,_, au + b , 

= — 7= cosh l -r (a positive, ac< ft 2 .) 

vo (b 2 -ac)* 

i •,,. _i_ *» 



1 _, au 4- 6 / x . v 

7== cos Ti; a* ( a ne 8 atlve -) 

—a (6 2 — acy 



v -a (ft* - acp 

17 e r du 1 aw -f b . . ., . 

175 - J(au 2 + 26u + c) = ^^ tan (^3^)1 <">*> 

= — ^tanh-i-^ ( ac<62 ' \ 

(ft 2 - ac)* (6 2 - ac)* Van + 6 < (ft 2 - ac)lj 

-_zL_oaai-*^±i r . ( ac<62 - ^ 

(ft 2 - ac)* (ft 2 - ac)* \ au + b > (ft 2 - ac)V 

176 . f <fe = _2 tanh -i J«E5 f 

«/ (a - tt) (u - 6)* (a - 6)* V a - 6 

* -2 . t /w-6 

or r tan -1 V r , 

(6 - a)* V & - a 



2 
or — 

(a 
2 



ay ■ u «• 

— t coth -1 y _ , . (The real form is to be taken.) 

177. f— life 2 ^J^ 

J (a - u) (6 - m)* (6 - a)* V 6 - a 

2 , . . . Ib-u 

or t coth * V t > 

(6 - a)* v ft- a 

or j tan -1 y r . (The real form is to be taken . ) 

178. T(w 2 - a 2 )* du = ^ u (u? - a 2 )*- ^ a 2 cosh" 1 -• 

179. f(a 2 - u 2 )* du = ^ u (a 2 - w 2 )* + J a 2 sin" 1 -• 

180. f(v? + a 2 )* du = I u (u 2 + a 2 )* + £ a 2 sinh" 1 -• 

J J JO 

182. ue au du = ^ (au - 1). 



APPENDIX 281 

/u m e" u m C 1 
u m e? u du = 1 / u m- V u du. 
a aj, 

r e° u du _ 1 |~_ _^_ P e? u du ] 

84, J vr "m-lL u*»- 1 + a J u*"- 1 ]' 

85. fa bu du = ^t 

J bloga 

/a u u n na u u n ~ x , n (n — 1) a u u n ~ 2 
log a (log a) 2 (log a) 3 

n (n - 1) (n -2 ) . . . 2-la» 

/Yr. u rlii. i 

87 



(loga) n+1 

log a (log a) 2 



/ardu _ a" 1 lo 

u n ~ n - 1 L W -1 in - 2) n"" 2 (n-2) (n-3) n"- 3 

(log a)"- 1 Ai tt dn ~] 

| " ' ' + (n-2) (n-3) . . . 2*1 J u \' 

00 Ca u du , . (m log a) 2 (n log a) 3 

88. J— = logn + nloga+-^| r + 3.3, + •••• 



89. Jjfe- log j 



dn 1 



90. C—ri— = —[mu- log (a + &e"«)]. 

91 . r — *• __* tw*fr-^). 

J ae™ + be-" 1 " m (^i V V ft/ 

92 - / / — Hi — a = 7= Dog (Va + be mu - Va) 

- log (Va+ 6e mu + Va)]. 
/■ jue»du_ e u 
9d ' J (l + n) 2 ~l + u' 



/• . , e° u logn 1 Pe au du 

94. le au logndn = 5 I 

J a aj u 

95. / log n dn = u log n — w. 

96. fu m log udu = u m+1 \^£ - , * J - 
J L™ + 1 (m + 1) 2 J 

97. / (log n) n dn = u (log w) n — n / (log n) n_1 dn. 

98. fu m (log n)» dn = um+l ^ u)n - — X7 /V (log u)*" 1 <*w. 
J v— » / m + 1 w + lj 

/ (log n) n dn _ (log n) n+l 
n n + 1 



282 APPENDIX 

m Ji— = log(logu) + logu + ^72! + 3T3! + "\ 

201 C du 5 _J_ r du t 

™ J (log u)» (n - 1) (log u)"" 1 T n-lJ (log u) n ~ l 

<xv> C u m du u m+l , m + 1 C u m du 

J (logu)" " (n - 1) (logw)"- 1 + n - 1 J (logii)— 1 ' 

203. / -j = I — dy, where y = — (m + 1) log m. 

204. f**_ = log (logu). 
J u log u 

205 f—^— = - 

J u (logM) n (n — 1) (logu) n_1 

206. f(a + bu) n log u du = s 

1 |\ , , wl , f(a + 6m)"» +1 oV | 

F^TTT) L (a + 6m) log u ~ J i J ' 

207. / u m log (a + 6m) du = 

_yy« * (.+*)- > /£*?]. 

20g Hog (a + 6m) <fa = 
J u 

. . . 6m 1 /6m\ 2 . 1 /buV 
loga.log W + --2H-)+3 2 (-] 

, 4 (log6w)2 _ 6 ± + l(^_l(«J + .... 

/Mog_udu_ i r iog^ , r rfM l 

J (a + &»)"• b (m - 1) L (a + few)"- 1 J m (a + 6m)"- 1 J ' 
om Tlogudtt 1, , , i , s 1 riog(a + 6w) . 

211. J {a + bu)\o g udu = ^^\o g u- a ^-au-\bu\ 

212. fJ^ wdw - 
J (a 



(a + 6m) * 



- Va log (Va + bu — Va)], if a > 0, 



^ [(log u - 2) V(a + 6m) + Valog (Va + 6m + Va) 



= ^[(logM-2) V(a+6M) +2 V^tan" 1 \/ ^rJ > if ° < °- 



APPENDIX 283 



213. r e-* 1 « t du = ^ = 7 }-r(l). 
Jo 2a 2a \2/ 

214. £ U -e-'du = T^- i 



l n + 1 rt»»+l 



a n 



215. f.-^,. 1 ' 3 ' 5 -,-' 2 "-" ^. 
Jo 2 n+1 a n V a 

217. r c-»" Vtt du « JL t/I. 

Jo 2 n V n 

218. r e ^-du = J-- (n>0.) 
Jo Vm v w 



>f 



219 



220 



sinh (nu) 2 n 
r°° udu t 2 






sinh (nw) 4 w 2 

Xv* nix 

sinh (raw) • sinh (nu) du = j cosh (raw) • cosh (nw) du 

= 0, if ra is different from n. 

cosh 2 (raw) du = — / sinh 2 (mu) du = -~ 

223. / sinh (raw) du = 0. 

J— Mr 

224. I cosh (m«) du = 0. 

225. / sinh (mu) cosh (nu) dw = 0. 
J— »*■ 

226. I sinh (raw) cosh (mu) du = 0. 

227. rMi dtt= _^. 

Jo 1 — u 6 

228. /"Mil <*„=-£. 

Jo 1 + m 12 



284 APPENDIX 

233. P(\ogu) n du= (-l)-.n!. 

886. jT-^U.v?. 

237. ^ tt-log (±)"du = ( ^ ( ^ ) !! 1 , if m + 1 > 0, n + 1 > 0. 



APPENDIX 



285 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 



X 


log cosh x 


log sinh x 


log tanh x 


, sinh x 
log 

X 


M#l iT~ 

tanh x 


0.000 


0.000 000 


-Inf. 


-Inf. 


0.000 000 


0.000 000 


0.001 


0.000 000 


7.000 000 


7.000 000 


0.000 000 


0.000 000 


0.002 


0.000 001 


7.301030 


7.301 029 


0.000 000 


0.000 001 


0.003 


0.000 002 


7.477 122 


7.477 120 


0.000 001 


0.000 001 


0.004 


0.000 003 


7.602 061 


7.602 058 


0.000 001 


0.000 002 


0.005 


0.000 005 


7.698 972 


7.698 966 


0.000 002 


0.000 004 


0.006 


0.000 008 


7.778 154 


7.778 146 


0.000 003 


0.000 005 


0.007 


0.000 011 


7.845 102 


7.845 091 


0.000 004 


0.000 007 


0.008 


0.000 014 


7.903 095 


7.903 081 


0.000 005 


0.000 009 


0.009 


0.000 018 


7.954 248 


7.954 231 


0.000 006 


0.000 012 


0.010 


0.000 022 


8.000 007 


7.999 986 


0.000 007 


0.000 014 


0.011 


0.000 026 


8.041401 


8.041 375 


0.000 009 


0.000 018 


0.012 


0.000 031 


8.079 192 


8.079 160 


0.000 010 


0.000 021 


0.013 


0.000 037 


8.113 956 


8.113 919 


0.000 012 


0.000 024 


0.014 


0.000 043 


8.146 142 


8.146 100 


0.000 014 


0.000 028 


0.015 


0.000 049 


8.176 108 


8.176 059 


0.000 016 


0.000 033 


0.016 


0.000 056 


8.204 139 


8.204 083 


0.000 019 


0.000 037 


0.017 


0.000 063 


8.230 470 


8.230 407 


0.000 021 


0.000 042 


0.018 


0.000 070 


8.255 296 


8.255 226 


0.000 023 


0.000 047 


0.019 


0.000 078 


8.278 780 


8.278 701 


0.000 026 


0.000 052 


0.020 


0.000 087 


8.301059 


8.300 972 


0.000 029 


0.000 058 


0.021 


0.000 096 


8.322 251 


8.322 155 


0.000 032 


0.000 064 


0.022 


0.000 105 


8.342 458 


8.342 353 


0.000 035 


0.000 070 


0.023 


0.000 115 


8.361 766 


8.361651 


0.000 038 


0.000 077 


0.024 


0.000 125 


8.380 253 


8.380 128 


0.000 042 


0.000 083 


0.025 


0.000 136 


8.397 985 


8.397 850 


0.000 045 


0.000 090 


0.026 


0.000 147 


8.415 022 


8.414 876 


0.000 049 


0.000 098 


0.027 


0.000 158 


8.431 417 


8.431 258 


0.000 053 


0.000 106 


0.028 


0.000 170 


8.447 215 


8.447 045 


0.000 057 


0.000 113 


0.029 


0.000 183 


8.462 459 


8.462 276 


0.000 061 


0.000 122 


0.030 


0.000 195 


8.477 186 


8.476 991 


0.000 065 


0.000 130 


0.031 


0.000 209 


8.491431 


8.491223 


0.000 070 


0.000 139 


0.032 


0.000 222 


8.505 224 


8.505 002 


0.000 074 


0.000 148 


0.033 


0.000 236 


8.518 593 


8.518 356 


0.000 079 


0.000 158 


0.034 


0.000 251 


8.531563 


8.531 312 


0.000 084 


0.000 167 


0.035 


0.000 266 


8.544 157 


8.543 891 


0.000 089 


0.000 177 


0.036 


0.000 281 


8.556 396 


8.556 115 


0.000 094 


0.000 188 


0.037 


0.000 297 


8.568 301 


8.568 004 


0.000 099 


0.000 198 


0.038 


0.000 313 


8.579 888 


8.579 575 


0.000 105 


0.000 209 


0.039 


0.000 330 


8.591 175 


8.590 844 


0.000 110 


0.000 220 


0.040 


0.000 347 


8.602 176 


8.601 828 


0.000 116 


0.000 232 


0.041 


0.000 365 


8.612 906 


8.612 541 


0.000 122 


0.000 243 


0.042 


0.000 383 


8.623 377 


8.622 994 


0.000 128 


0.000 255 


0.043 


0.000 401 


8.633 602 


8.633 201 


0.000 134 


0.000 268 


0.044 


0.000 420 


8.643 593 


8.643 173 


0.000 140 


0.000 280 


0.045 


0.000 440 


8.653 359 


8.652 920 


0.000 147 


0.000 293 


0.046 


0.000 459 


8.662 911 


8.662 452 


0.000 153 


0.000 306 


0.047 


0.000 480 


8.672 258 


8.671 778 


0.000 160 


0.000 320 


0.048 


0.000 500 


8.681408 


8.680 908 


0.000 167 


0.000 333 


0.049 


0.000 521 


8.690 370 


8.689 849 


0.000 174 


0.000 347 



286 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh x 


log sinh x 


log tanh x 


, sinh x 
log 

X 


tanhz 


0.050 


0.000 543 


8.699 151 


8.698 608 


0.000 181 


0.000 362 


0.051 


0.000 565 


8.707 758 


8.707 194 


0.000 188 


0.000 376 


0.052 


0.000 587 


8.716 199 


8.715 612 


0.000 196 


0.000 391 


0.053 


0.000 610 


8.724 479 


8.723 869 


0.000 203 


0.000 406 


0.054 


0.000 633 


8.732 605 


8.731972 


0.000 211 


0.000 422 


0.055 


0.000 657 


8.740 582 


8.739 925 


0.000 219 


0.000 438 


0.056 


0.000 681 


8.748 415 


8.747 734 


0.000 227 


0.000 454 


0.057 


0.000 705 


8.756 110 


8.755 405 


0.000 235 


0.000 470 


0.058 


0.000 730 


8.763 671 


8.762 941 


0.000 243 


0.000 487 


0.059 


0.000 755 


8.771 104 


8.770 348 


0.000 252 


0.000 504 


0.060 


0.000 781 


8.778 412 


8.777 631 


0.000 261 


0.000 521 


0.061 


0.000 808 


8.785 599 


8.784 792 


0.000 269 


0.000 538 


0.062 


0.000 834 


8.792 670 


8.791836 


0.000 278 


0.000 556 


0.063 


0.000 861 


8.799 628 


8.798 767 


0.000 287 


0.000 574 


0.064 


0.000 889 


8.806 476 


8.805 588 


0.000 296 


0.000 592 


0.065 


0.000 917 


8.813 219 


8.812 302 


0.000 306 


0.000 611 


0.066 


0.000 945 


8.819 859 


8.818 914 


0.000 315 


0.000 630 


0.067 


0.000 974 


8.826 400 


8.825 426 


0.000 325 


0.000 649 


0.068 


0.001 003 


8.832 844 


8.831840 


0.000 335 


0.000 669 


0.069 


0.001 033 


8.839 194 


8.838 161 


0.000 345 


0.000 688 


0.070 


0.001063 


8.845 453 


8.844 390 


0.000 355 


0.000 709 


0.071 


0.001 094 


8.851 623 


8.850 529 


0.000 365 


0.000 729 


0.072 


0.001 125 


8.857 708 


8.856 583 


0.000 375 


0.000 750 


0.073 


0.001 156 


8.863 709 


8.862 552 


0.000 386 


0.000 770 


0.074 


0.001 188 


8.869 628 


8.868 440 


0.000 396 


0.000 792 


0.075 


0.001 220 


8.875 468 


8.874 248 


0.000 407 


0.000 813 


0.076 


0.001 253 


8.881 232 


8.879 979 


0.000 418 


0.000 835 


0.077 


0.001286 


8.886 920 


8.885 634 


0.000 429 


0.000 857 


0.078 


0.001320 


8.892 535 


8.891 215 


0.000 440 


0.000 880 


0.079 


0.001 354 


8.898 079 


8.896 725 


0.000 452 


0.000 902 


0.080 


0.001 388 


8.903 553 


8.902 165 


0.000 463 


0.000 925 


0.081 


0.001 423 


8.908 960 


8.907 537 


0.000 475 


0.000 948 


0.082 


0.001 458 


8.914 300 


8.912 842 


0.000 487 


0.000 972 


0.083 


0.001 494 


8.919 577 


8.918 082 


0.000 499 


0.000 996 


0.084 


0.001 530 


8.924 790 


8.923 260 


0.000 511 


0.001 020 


0.085 


0.001 567 


8.929 942 


8.928 375 


0.000 523 


0.001044 


0.086 


0.001604 


8.935 034 


8.933 430 


0.000 535 


0.001 069 


0.087 


0.001642 


8.940 067 


8.938 425 


0.000 548 


0.001 094 


0.088 


0.001 679 


8.945 043 


8.943 364 


0.000 560 


0.001 119 


0.089 


0.001 718 


8.949 963 


8.948 245 


0.000 573 


0.001 145 


0.090 


0.001 757 


8.954 829 


8.953 072 


0.000 586 


0.001 170 


0.091 


0.001 796 


8.959 641 


8.957 845 


0.000 599 


0.001 196 


0.092 


0.001835 


8.964 400 


8.962 565 


0.000 612 


0.001 223 


0.093 


0.001 875 


8.969 109 


8.967 233 


0.000 626 


0.001250 


0.094 


0.001 916 


8.973 767 


8.971 851 


0.000 639 


0.001 277 


0.095 


0.001 957 


8.978 377 


8.976 420 


0.000 653 


0.001 304 


0.096 


0.001998 


8.982 938 


8.980 940 


0.000 667 


0.001 331 


0.097 


0.002 040 


8.987 453 


8.985 413 


0.000 681 


0.001 359 


0.098 


0.002 082 


8.991 921 


8.989 839 


0.000 695 


0.001387 


0.099 


0.002 125 


8.996 344 


8.994 220 


0.000 709 


0.001 416 



APPENDIX 



287 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh x 


log sinh x 


log tanh x 


. sinh x 
log — 


log, x 
tanhz 


0.100 


0.002 168 


9.000 724 


8.998 556 


0.000 724 


0.001444 


0.101 


0.002 211 


9.005 059 


9.002 848 


0.000 738 


0.001473 


0.102 


0.002 255 


9.009 353 


9.007 098 


0.000 753 


0.001502 


0.103 


0.002 300 


9.013 605 


9.011 305 


0.000 768 


0.001532 


0.104 


0.002 344 


9.017 816 


9.015 472 


0.000 783 


0.001 562 


0.105 


0.002 390 


9.021987 


9.019 597 


0.000 798 


0.001592 


0.106 


0.002 435 


9.026 119 


9.023 684 


0.000 813 


0.001622 


0.107 


0.002 481 


9.030 212 


9.027 731 


0.000 828 


0.001653 


0.108 


0.002 528 


9.034 268 


9.031 740 


0.000 844 


0.001 684 


0.109 


0.002 575 


9.038 286 


9.035 711 


0.000 860 


0.001 715 


0.110 


0.002 622 


9.042 268 


9.039 646 


0.000 875 


0.001 747 


0.111 


0.002 670 


9.046 214 


9.043 544 


0.000 891 


0.001 779 


0.112 


0.002 718 


9.050126 


9.047 407 


0.000 908 


0.001811 


0.113 


0.002 767 


9.054 002 


9.051235 


0.000 924 


0.001 843 


0.114 


0.002 816 


9.057 845 


9.055 029 


0.000 940 


0.001 876 


0.115 


0.002 865 


9.061 655 


9.058 789 


0.000 957 


0.001909 


0.116 


0.002 915 


9.065 432 


9.062 516 


0.000 974 


0.001 942 


0.117 


0.002 966 


9.069 176 


9.066 210 


0.000 990 


0.001975 


0.118 


0.003 017 


9.072 889 


9.069 873 


0.001007 


0.002 009 


0.119 


0.003 068 


9.076 571 


9.073 504 


0.001025 


0.002 043 


0.120 


0.003 119 


9.080 223 


9.077 104 


0.001042 


0.002 078 


0.121 


0.003 172 


9.083 845 


9.080 673 


0.001059 


0.002 112 


0.122 


0.003 224 


9.087 437 


9.084 213 


0.001 077 


0.002 147 


0.123 


0.003 277 


9.091000 


9.087 723 


0.001 095 


0.002 182 


0.124 


0.003 330 


9.094 534 


9.091 204 


0.001 112 


0.002 218 


0.125 


0.003 384 


9.098 040 


9.094 656 


0.001 130 


0.002 254 


0.126 


0.003 438 


9.101519 


9.098 081 


0.001 149 


0.002 290 


0.127 


0.003 493 


9.104 971 


9.101478 


0.001 167 


0.002 326 


0.128 


0.003 548 


9.108 395 


9.104 847 


0.001 185 


0.002 363 


0.129 


0.003 604 


9.111794 


9.108190 


0.001204 


0.002 400 


0.130 


0.003 659 


9.115 166 


9.111506 


0.001223 


0.002 437 


0.131 


0.003 716 


9.118 513 


9.114 797 


0.001241 


0.002 474 


0.132 


0.003 773 


9.121834 


9.118 062 


0.001 260 


0.002 512 


0.133 


0.003 830 


9.125 131 


9.121301 


0.001280 


•0.002 550 


0.134 


0.003 887 


9.128 404 


9.124 516 


0.001299 


0.002 589 


0.135 


0.003 946 


9.131652 


9.127 707 


0.001 318 


0.002 627 


0.136 


0.004 004 


9.134 877 


9.130 873 


0.001338 


0.002 666 


0.137 


0.004 063 


9.138 078 


9.134 015 


0.001358 


0.002 705 


0.138 


0.004 122 


9.141257 


9.137 134 


0.001 378 


0.002 745 


0.139 


0.004 182 


9.144 412 


9.140 230 


0.001 398 


0.002 784 


0.140 


0.004 242 


9.147 546 


9.143 304 


0.001 418 


0.002 824 


0.141 


0.004 303 


9.150 657 


9.146 354 


0.001438 


0.002 865 


0.142 


0.004 364 


9.153 747 


9.149 383 


0.001 459 


0.002 905 


0.143 


0.004 425 


9.156 815 


9.152 390 


0.001 479 


0.002 946 


0.144 


0.004 487 


9.159 862 


9.155 375 


0.001500 


0.002 987 


0.145 


0.004 550 


9.162 889 


9.158 339 


0.001 521 


0.003 029 


0.146 


0.004 612 


9.165 895 


9.161282 


0.001542 


0.003 071 


0.147 


0.004 676 


9.168 880 


9.164 205 


0.001563 


0.003 113 


0.148 


0.004 739 


9.171846 


9.167 107 


0.001584 


0.003 155 


0.149 


0.004 803 


9.174 792 


9.169 989 


0.001606 


0.003 197 



2SS 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh z 


log sinh x 


log tanh x 


sinh x 
log 


,08 tanhx 


0.150 


0.004 868 


9.177 719 


9.172 851 


0.001 627 


0.003 240 


0.151 


0.004 932 


9.180 626 


9.175 694 


0.001 649 


0.003 283 


0.152 


0.004 998 


9.183 515 


9.178 517 


0.001 671 


0.003 327 


0.153 


0.005 063 


9.186 385 


9.181321 


0.001 693 


0.003 370 


0.154 


0.005 130 


9.189 236 


9.184 106 


0.001 715 


0.003 414 


0.155 


0.005 196 


9.192 069 


9.186 873 


0.001 738 


0.003 459 


0.156 


0.005 263 


9.194 885 


9.189 621 


0.001760 


0.003 503 


0.157 


0.005 331 


9.197 682 


9.192 352 


0.001 783 


0.003 548 


0.158 


0.005 398 


9.200 463 


9.195 064 


0.001 805 


0.003 593 


0.159 


0.005 467 


9.203 225 


9.197 759 


0.001 828 


0.003 638 


0.160 


0.005 535 


9.205 971 


9.200 436 


0.001851 


0.003 684 


0.161 


0.005 605 


9.208 700 


9.203 096 


0.001 875 


0.003 730 


0.162 


0.005 674 


9.211413 


9.205 739 


0.001898 


0.003 776 


0.163 


0.005 744 


9.214 109 


9.208 365 


0.001 921 


0.003 823 


0.164 


0.005 814 


9.216 789 


9.210 975 


0.001945 


0.003 869 


0.165 


0.005 885 


9.219 453 


9.213 568 


0.001 969 


0.003 916 


0.166 


0.005 956 


9.222 101 


9.216 144 


0.001 993 


0.003 964 


0.167 


0.006 028 


9.224 733 


9.218 705 


0.002 017 


0.004 011 


0.168 


0.006 100 


9.227 350 


9.221 250 


0.002 041 


0.004 059 


0.169 


0.006 173 


9.229 952 


9.223 779 


0.002 065 


0.004 107 


0.170 


0.006 246 


9.232 539 


9.226 293 


0.002 090 


0.004156 


0.171 


0.006 319 


9.235 111 


9.228 792 


0.002 114 


0.004 204 


0.172 


0.006 393 


9.237 668 


9.231 275 


0.002 139 


0.004 253 


0.173 


0.006 467 


9.240 210 


9.233 743 


0.002 164 


0.004 303 


0.174 


0.006 541 


9.242 738 


9.236 197 


0.002 189 


0.004 352 


0.175 


0.006 616 


9.245 253 


9.238 636 


0.002 214 


0.004 402 


0.176 


0.006 692 


9.247 752 


9.241 061 


0.002 240 


0.004 452 


0.177 


0.006 768 


9.250 239 


9.243 471 


0.002 265 


0.004 502 


0.178 


0.006 844 


9.252 711 


9.245 867 


0.002 291 


0.004 553 


0.179 


0.006 921 


9.255 170 


9.248 249 


0.002 317 


0.004 604 


0.180 


0.006 998 


9.257 615 


9.250 617 


0.002 343 


0.004 655 


0.181 


0.007 075 


9.260 047 


9.252 972 


0.002 369 


0.004 707 


0.182 


0.007 153 


9.262 466 


9.255 313 


0.002 395 


0.004 758 


0.183 


0.007 232 


9.264 872 


9.257 641 


0.002 421 


0.004 810 


0.184 


0.007 311 


9.267 266 


9.259 955 


0.002 448 


0.004 863 


0.185 


0.007 390 


9.269 646 


9.262 256 


0.002 474 


0.004 915 


0.186 


0.007 470 


9.272 014 


9.264 545 


0.002 501 


0.004 968 


0.187 


0.007 550 


9.274 370 


9.266 820 


0.002 528 


0.005 021 


0.188 


0.007 630 


9.276 713 


9.269 083 


0.002 555 


0.005 075 


0.189 


0.007 711 


9.279 044 


9.271333 


0.002 583 


0.005 128 


0.190 


0.007 792 


9.281 363 


9.273 571 


0.002 610 


0.005 182 


0.191 


0.007 874 


9.283 671 


9.275 797 


0.002 637 


0.005 237 


0.192 


0.007 956 


9.285 966 


9.278 010 


0.002 665 


0.005 291 


0.193 


0.008 039 


9.288 250 


9.280 211 


0.002 693 


0.005 346 


0.194 


0.008 122 


9.290 523 


9.282 401 


0.002 721 


0.005 401 


0.195 


0.008 205 


9.292 783 


9.284 578 


0.002 749 


0.005 456 


0.196 


0.008 289 


9.295 033 


9.286 744 


0.002 777 


0.005 512 


0.197 


0.008 373 


9.297 272 


9.288 898 


0.002 805 


0.005 568 


0.198 


0.008 458 


9.299 499 


9.291041 


0.002 834 


0.005 624 


0.199 


0.008 543 


9.301 716 


9.293 173 


0.002 863 


0.005 680 



APPENDIX 



289 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 





log cosh x 


log ainh x 


log tanh x 


, sinh x 
log 

X 


X 


X 


tanh i 


0.200 


0.008 629 


9.303 921 


9.295 293 


0.002 891 


0.005 737 


0.201 


0.008 715 


9.306 116 


9.297 402 


0.002 920 


0.005 794 


0.202 


0.008 801 


9.308 301 


9.299 500 


0.002 949 


0.005 851 


0.203 


0.008 888 


9.310 475 


9.301587 


0.002 979 


0.005 909 


0.204 


0.008 975 


9.312 638 


9.303 663 


0.003 008 


0.005 967 


0.205 


0.009 062 


9.314 791 


9.305 729 


0.003 038 


0.006 025 


0.206 


0.009 150 


9.316 935 


9.307 784 


0.003 067 


0.006 083 


0.207 


0.009 239 


9.319 067 


9.309 829 


0.003 097 


0.006 142 


0.208 


0.009 328 


9.321 190 


9.311 863 


0.003 127 


0.006 201 


0.209 


0.009 417 


9.323 303 


9.313 886 


0.003 157 


0.006 260 


0.210 


0.009 507 


9.325 407 


9.315 900 


0.003 187 


0.006 319 


0.211 


0.009 597 


9.327 500 


9.317 904 


0.003 218 


0.006 379 


0.212 


0.009 687 


9.329 584 


9.319 897 


0.003 248 


0.006 439 


0.213 


0.009 778 


9.331 659 


9.321 880 


0.003 279 


0.006 499 


0.214 


0.009 869 


9.333 724 


9.323 854 


0.003 310 


0.006 560 


0.215 


0.009 961 


9.335 779 


9.325 818 


0.003 341 


0.006 621 


0.216 


0.010 053 


9.337 826 


9.327 772 


0.003 372 


0.006 682 


0.217 


0.010 146 


9.339 863 


9.329 717 


0.003 403 


0.006 743 


0.218 


0.010 239 


9.341 891 


9.331652 


0.003 434 


0.006 805 


0.219 


0.010 332 


9.343 910 


9.333 578 


0.003 466 


0.006 866 


0.220 


0.010 426 


9.345 920 


9.335 494 


0.003 498 


0.006 929 


0.221 


0.010 520 


9.347 922 


9.337 401 


0.003 529 


0.006 991 


0.222 


0.010 615 


9.349 914 


9.339 299 


0.003 561 


0.007 054 


0.223 


0.010 710 


9.351898 


9.341 188 


0.003 594 


0.007 117 


0.224 


0.010 806 


9.353 874 


9.343 068 


0.003 626 


0.007 180 


0.225 


0.010 902 


9.355 841 


9.344 939 


0.003 658 


0.007 243 


0.226 


0.010 998 


9.357 799 


9.346 801 


0.003 691 


0.007 307 


0.227 


0.011095 


9.359 749 


9.348 655 


0.003 723 


0.007 371 


0.228 


0.011 192 


9.361 691 


9.350 499 


0.003 756 


0.007 435 


0.229 


0.011 289 


9.363 625 


9.352 335 


0.003 789 


0.007 500 


0.230 


0.011387 


9.365 550 


9.354 163 


0.003 822 


0.007 565 


0.231 


0.011 486 


9.367 468 


9.355 982 


0.003 856 


0.007 630 


0.232 


0.011 584 


9.369 377 


9.357 793 


0.003 889 


0.007 695 


0.233 


0.011 684 


9.371 278 


9.359 595 


0.003 922 


0.007 761 


0.234 


0.011783 


9.373 172 


9.361 389 


0.003 956 


0.007 827 


0.235 


0.011883 


9.375 058 


9.363 175 


0.003 990 


0.007 893 


0.236 


0.011984 


9.376 936 


9.364 952 


0.004 024 


0.007 960 


0.237 


0.012 084 


9.378 806 


9.366 722 


0.004 058 


0.008 026 


0.238 


0.012 186 


9.380 669 


9.368 484 


0.004 092 


0.008 093 


0.239 


0.012 287 


9.382 525 


,9.370 237 


0.004 127 


0.008 161 


0.240 


0.012 389 


9.384 372 


9.371983 


0.004 161 


0.008 228 


0.241 


0.012 492 


9.386 213 


9.373 721 


0.004 196 


0.008 296 


0.242 


0.012 595 


9.388 046 


9.375 451 


0.004 231 ■ 


0.008 364 


0.243 


0.012 698 


9.389 872 


9.377 174 


0.004 266 


0.008 432 


0.244 


0.012 802 


9.391 691 


9.378 889 


0.004 301 


0.008 501 


0.245 


0.012 906 


9.393 502 


9.380 596 


0.004 336 


0.008 570 


0.246 


0.013 010 


9.395 307 


9.382 296 


0.004 371 


0.008 639 


0.247 


0.013 115 


9.397 104 


9.383 989 


0.004 407 


0.008 708 


0.248 


0.013 221 


9.398 894 


9.385 674 


0.004 443 


0.008 778 


0.249 


0.013 326 


9.400 678 


9.387 351 


0.004 479 


0.008 848 



200 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log ooeh x 


log sinh x 


log tanh z 


. sinh z 
log 


log: — c - 
tanh i 


0.250 


0.013 433 


9.402 455 


9.389 022 


0.004 515 


0.008 918 


0.251 


0.013 539 


9.404 224 


9.390 685 


0.004 551 


0.008 989 


0.252 


0.013 646 


9.405 987 


9.392 341 


0.004 587 


0.009 059 


0.253 


0.013 754 


9.407 744 


9.393 990 


0.004 623 


0.009 130 


0.254 


0.013 861 


9.409 494 


9.395 632 


0.004 660 


0.009 202 


0.255 


0.013 970 


9.411 237 


9.397 267 


0.004 697 


0.009 273 


0.256 


0.014 078 


9.412 973 


9.398 895 


0.004 733 


0.009 345 


0.257 


0.014 187 


9.414 703 


9.400 516 


0.004 770 


0.009 417 


0.258 


0.014 297 


9.416 427 


9.402 130 


0.004 807 


0.009 489 


0.259 


0.014 406 


9.418 144 


9.403 738 


0.004 845 


0.009 562 


0.260 


0.014 517 


9.419 855 


9.405 339 


0.004 882 


0.009 635 


0.261 


0.014 627 


9.421560 


9.406 933 


0.004 920 


0.009 708 


0.262 


0.014 738 


9.423 259 


9.408 520 


0.004 957 


0.009 781 


0.263 


0.014 850 


9.424 951 


9.410 101 


0.004 995 


0.009 855 


0.264 


0.014 962 


9.426 637 


9.411 675 


0.005 033 


0.009 929 


0.265 


0.015 074 


9.428 317 


9.413 243 


0.005 071 


0.010 003 


0.266 


0.015 187 


9.429 991 


9.414 804 


0.005 109 


0.010 077 


0.267 


0.015 300 


9.431 659 


9.416 359 


0.005 148 


0.010 152 


0.268 


0.015 413 


9.433 321 


9.417 908 


0.005 186 


0.010 227 


0.269 


0.015 527 


9.434 977 


9.419 450 


0.005 225 


0.010 302 


0.270 


0.015 641 


9.436 628 


9.420 986 


0.005 264 


0.010 377 


0.271 


0.015 756 


9.438 272 


9.422 516 


0.005 303 


0.010 453 


0.272 


0.015 871 


9.439 911 


9.424 040 


0.005 342 


0.010 529 


0.273 


0.015 987 


9.441544 


9.425 557 


0.005 381 


0.010 605 


0.274 


0.016 103 


9.443 171 


9.427 069 


0.005 421 


0.010 682 


0.275 


0.016 219 


9.444 793 


9.428 574 


0.005 460 


0.010 759 


0.276 


0.016 336 


9.446 409 


9.430 073 


0.005 500 


0.010 836 


0.277 


0.016 453 


9.448 019 


9.431567 


0.005 540 


0.010 913 


0.278 


0.016 570 


9.449 624 


9.433 054 


0.005 580 


0.010 991 


0.279 


0.016 688 


9.451224 


9.434 536 


0.005 620 


0.011068 


0.280 


0.016 806 


9.452 818 


9.436 012 


0.005 660 


0.011 146 


0.281 


0.016 925 


9.454 407 


9.437 482 


0.005 700 


0.011225 


0.282 


0.017 044 


9.455 990 


9.438 946 


0.005 741 


0.011 303 


0.283 


0.017 164 


9.457 568 


9.440 404 


0.005 782 


0.011382 


0.284 


0.017 284 


9.459 141 


9.441 857 


0.005 822 


0.011461 


0.285 


0.017 404 


9.460 708 


9.443 304 


0.005 863 


0.011541 


0.286 


0.017 525 


9.462 271 


9.444 746 


0.005 905 


0.011 620 


0.287 


0.017 646 


9.463 828 


9.446 182 


0.005 946 


0.011 700 


0.288 


0.017 767 


9.465 380 


9.447 612 


0.005 987 


0.011 780 


0.289 


0.017 889 


9.466 927 


9.449 037 


0.006 029 


0.011 861 


0.290 


0.018 012 


9.468 468 


9.450 457 


0.006 070 


0.011 941 


0.291 


0.018 134 


9.470 005 


9.451 871 


0.006 112 


0.012 022 


0.292 


0.018 258 


9.471 537 


9.453 279 


0.006 154 


0.012 103 


0.293 


0.018 381 


9.473 064 


9.454 683 


0.006 196 


0.012 185 


0.294 


0.018 505 


9.474 586 


9.456 081 


0.006 239 


0.012 267 


0.295 


0.018 629- 


9.476 103 


9.457 474 


0.006 281 


0.012 348 


0.296 


0.018 754 


9.477 615 


9.458 861 


0.006 323 


0.012 431 


0.297 


0.018 879 


9.479 123 


9.460 243 


0.006 366 


0.012 513 


0.298 


0.019 005 


9.480 625 


9.461 620 


0.006 409 


0.012 596 


. 0.299 


0.019 131 


9.482 123 


9.462 992 


0.006 452 


0.012 679 



APPENDIX 



291 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh z 


log sinh x 


log tanh x 


. sinh x 
log 


lo 8; — j— 

tanh x 


0.300 


0.019 257 


9.483 616 


9.464 359 


0.006 495 


0.012 762 


0.301 


0.019 384 


9.485 105 


9.465 721 


0.006 538 


0.012 845 


0.302 


0.019 511 


9.486 589 


9.467 078 


0.006 582 


0.012 929 


0.303 


0.019 638 


9.488 068 


9.468 429 


0.006 625 


0.013 013 


0.304 


0.019 766 


9.489 542 


9.469 776 


0.006 669 


0.013 097 


0.305 


0.019 894 


9.491 012 


9.471 118 


0.006 713 


0.013 182 


0.306 


0.020 023 


9.492 478 


9.472 455 


0.006 757 


0.013 267 


0.307 


0.020 152 


9.493 939 


9.473 787 


0.006 801 


0.013 352 


0.308 


0.020 282 


9.495 396 


9.475 114 


0.006 845 


0.013 437 


0.309 


0.020 412 


9.496 848 


9.476 436 


0.006 889 


0.013 522 


0.310 


0.020 542 


9.498 295 


9.477 754 


0.006 934 


0.013 608 


0.311 


0.020 673 


9.499 739 


9.479 066 


0.006 978 


0.013 694 


0.312 


0.020 804 


9.501 178 


9.480 374 


0.007 023 


0.013 780 


0.313 


0.020 935 


9.502 613 


9.481 677 


0.007 068 


0.013 867 


0.314 


0.021 067 


9.504 043 


9.482 976 


0.007 113 


0.013 954 


0.315 


0.021 199 


9.505 469 


9.484 270 


0.007 159 


0.014 041 


0.316 


0.021 332 


9.506 891 


9.485 559 


0.007 204 


0.014 128 


0.317 


0.021 465 


9.508 309 


9.486 844 


0.007 249 


0.014 216 


0.318 


0.021 598 


9.509 722 


9.488 124 


0.007 295 


0.014 303 


0.319 


0.021 732 


9.511 132 


9.489 399 


0.007 341 


0.014 391 


0.320 


0.021 866 


9.512 537 


9.490 670 


0.007 387 


0.014 480 


0.321 


0.022 001 


9.513 938 


9.491 937 


0.007 433 


0.014 568 


0.322 


0.022 136 


9.515 335 


9.493 199 


0.007 479 


0.014 657 


0.323 


0.022 271 


9.516 728 


9.494 457 


0.007 525 


0.014 746 


0.324 


0.022 407 


9.518117 


9.495 710 


0.007 572 


0.014 835 


0.325 


0.022 543 


9.519 502 


9.496 959 


0.007 619 


0.014 925 


0.326 


0.022 680 


9.520 883 


9.498 203 


0.007 665 


0.015 015 


0.327 


0.022 817 


9.522 260 


9.499 443 


0.007 712 


0.015 105 


0.328 


0.022 954 


9.523 633 


9.500 679 


0.007 759 


0.015 195 


0.329 


0.023 092 


9.525 003 


9.501911 


0.007 807 


0.015 285 


0.330 


0.023 230 


9.526 368 


9.503 138 


0.007 854 


0.015 376 


0.331 


0.023 369 


9.527 730 


9.504 361 


0.007 902 


0.015 467 


0.332 


0.023 508 


9.529 087 


9.505 580 


0.007 949 


0.015 558 


0.333 


0.023 647 


9.530 441 


9.506 794 


0.007 997 


0.015 650 


0.334 


0.023 787 


9.531 791 


9.508 005 


0.008 045 


0.015 742 


0.335 


0.023 927 


9.533 138 


9.509 211 


0.008 093 


0.015 834 


0.336 


0.024 067 


9.534 480 


9.510 413 


0.008 141 


0.015 926 


0.337 


0.024 208 


9.535 819 


9.511 611 


0.008 190 


0.016 019 


0.338 


0.024 349 


9.537 155 


9.512 805 


0.008 238 


0.016 111 


0.339 


0.024 491 


9.538 486 


9.513 995 


0.008 287 


0.016 204 


0.340 


0.024 633 


9.539 814 


9.515 181 


0.008 335 


0.016 298 


0.341 


0.024 775 


9.541 139 


9.516 363 


0.008 384 


0.016 391 


0.342 


0.024 918 


9.542 459 


9.517 541 


0.008 433 


0.016 485 


0.343 


0.025 061 


9.543 777 


9.518 715 


0.008 483 


0.016 579 


0.344 


0.025 205 


9.545 090 


9.519 885 


0.008 532 


0.016 673 


0.345 


0.025 349 


9.546 400 


9.521 052 


0.008 581 


0.016 768 


0.346 


0.025 493 


9.547 707 


9.522 214 


0.008 631 


0.016 862 


0.347 


0.025 638 


9.549 010 


9.523 372 


0.008 681 


0.016 957 


0.348 


0.025 783 


9.550 310 


9.524 527 


0.008 731 


0.017 052 


0.349 


0.025 929 


9.551 606 


9.525 678 


0.008 781 


0.017 148 



202 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh z 


log sinh z 


log tan h z 


, sinh z 
log 


log; T— 


0.350 


0.026 075 


9.552 899 


9.526 824 


0.008 831 


0.017 244 


0.351 


0.026 221 


9.554 188 


9.527 968 


0.008 881 


0.017 340 


0.352 


026 367 


9.555 474 


9.529 107 


0.008 932 


0.017 436 


0.353 


0.026 515 


9.556 757 


9.530 242 


0.008 982 


0.017 532 


0.354 


0.026 662 


9.558 036 


9.531 374 


0.009 033 


0.017 629 


0.355 


0.026 810 


9.559 312 


9.532 503 


0.009 084 


0.017 726 


0.356 


0.026 958 


9.560 585 


9.533 627 


0.009 135 


0.017 823 


0.357 


0.027 107 


9.561854 


9.534 748 


0.009 186 


0.017 920 


0.358 


0.027 256 


9.563 121 


9.535 865 


0.009 238 


0.018 018 


0.359 


0.027 405 


9.564 383 


9.536 979 


0.009 289 


0.018 116 


0.360 


0.027 555 


9.565 643 


9.538 088 


0.009 341 


0.018 214 


0.361 


0.027 705 


9.566 900 


9.539 195 


0.009 392 


0.018 312 


0.362 


0.027 855 


9.568 153 


9.540 298 


0.009 444 


0.018 411 


0.363 


0.028 006 


9.569 403 


9.541 397 


0.009 496 


0.018 510 


0.364 


0.028 157 


9.570 650 


9.542 492 


0.009 548 


0.018 609 


0.365 


0.028 309 


9.571894 


9.543 585 


0.009 601 


0.018 708 


0.366 


0.028 461 


9.573 134 


9.544 673 


0.009 653 


0.018 808 


0.367 


0.028 613 


9.574 372 


9.545 758 


0.009 706 


0.018 908 


0.368 


0.028 766 


9.575 606 


9.546 840 


0.009 758 


0.019 008 


0.369 


0.028 919 


9.576 838 


9.547 918 


0.009 811 


0.019 108 


0.370 


0.029 073 


9.578 066 


9.548 993 


0.009 864 


0.019 209 


0.371 


0.029 227 


9.579 291 


9.550 065 


0.009 917 


019 309 


0.372 


0.029 381 


9.580 514 


9.551 133 


0.009 971 


0.019 410 


0.373 


0.029 536 


9.581 733 


9.552 197 


0.010 024 


0.019 512 


0.374 


0.029 691 


9.582 949 


9.553 258 


0.010 078 


0.019 613 


0.375 


0.029 846 


9.584 163 


9.554 316 


0.010 131 


0.019 715 


0.376 


0.030 002 


9.585 373 


9.555 371 


0.010 185 


0.019 817 


0.377 


0.030 158 


9.586 581 


9.556 422 


0.010 239 


0.019 919 


0.378 


0.030 315 


9.587 785 


9.557 470 


0.010 293 


0.020 022 


0.379 


0.030 472 


9.588 987 


9.558 515 


0.010 348 


0.020 124 


0.380 


0.030 629 


9.590 186 


9.559 556 


0.010 402 


0.020 227 


0.381 


0.030 787 


9.591382 


9.560 595 


0.010 457 


0.020 330 


0.382 


0.030 945 


9.592 575 


9.561 630 


0.010 511 


0.020 434 


0.383 


0.031 104 


9.593 765 


9.562 662 


0.010 566 


0.020 537 


0.384 


0.031 262 


9.594 952 


9.563 690 


0.010 621 


0.020 641 


0.385 


0.031422 


9.596 137 


9.564 716 


0.010 676 


0.020 745 


0.386 


0.031581 


9.597 319 


9.565 738 


0.010 732 


0.020 850 


0.387 


0.031741 


9.598 498 


9.566 757 


0.010 787 


0.020 954 


0.388 


0.031 901 


9.599 674 


9.567 773 


0.010 843 


0.021059 


0.389 


0.032 062 


9.600 848 


9.568 786 


0.010 898 


0.021 164 


0.390 


0.032 223 


9.602 019 


9.569 795 


0.010 954 


0.021 269 


0.391 


0.032 385 


9.603 187 


9.570 802 


0.011010 


0.021 375 


0.392 


0.032 547 


9.604 352 


9.571 806 


0.011066 


0.021480 


0.393 


0.032 709 


9.605 515 


9.572 806 


0.011 122 


0.021586 


0.394 


0.032 871 


9.606 675 


9.573 804 


0.011 179 


0.021 693 


0.395 


0.033 034 


9.607 832 


9.574 798 


0.011 235 


0.021 799 


0.396 


0.033 198 


9.608 987 


9.575 789 


0.011292 


0.021906 


0.397 


0.033 361 


9.610 139 


9.576 778 


0.011 349 


0.022 013 


0.398 


0.033 525 


9.611 289 


9.577 763 


0.011406 


0.022 120 


0.399 


0.033 690 


9.612 436 


9.578 746 


0.011463 


0.022 227 



APPENDIX 



293 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh x 


log sinh x 


log tanh x 


. sinh x 
log 

X 


log. ' 
tanh x 


0.400 


0.033 855 


9.613 580 


9.579 725 


0.011 520 


0.022 335 


0.401 


0.034 020 


9.614 722 


9.580 702 


0.011 577 


0.022 443 


0.402 


0.034 186 


9.615 861 


9.581 675 


0.011 635 


0.022 551 


0.403 


0.034 351 


9.616 998 


9.582 646 


0.011 693 


0.022 659 


0.404 


0.034 518 


9.618 132 


9.583 614 


0.011 750 


0.022 767 


0.405 


0.034 684 


9.619 263 


9.584 579 


0.011 808 


0.022 876 


0.406 


0.034 852 


9.620 392 


9.585 541 


0.011 866 


0.022 985 


0.407 


0.035 019 


9.621519 


9.586 500 


0.011925 


0.023 094 


0.408 


0.035 187 


9.622 643 


9.587 456 


0.011 983 


0.023 204 


0.409 


0.035 355 


9.623 765 


9.588 410 


0.012 041 


0.023 313 


0.410 


0.035 523 


9.624 884 


9.589 360 


0.012 100 


0.023 423 


0.411 


0.035 692 


9.626 001 


9.590 308 


0.012 159 


0.023 534 


0.412 


0.035 862 


9.627 115 


9.591 253 


0.012 218 


0.023 644 


0.413 


0.036 031 


9.628 227 


9.592 196 


0.012 277 


0.023 754 


0.414 


0.036 201 


9.629 336 


9.593 135 


0.012 336 


0.023 865 


0.415 


0.036 372 


9.630 443 


9.594 072 


0.012 395 


0.023 976 


0.416 


0.036 542 


9.631548 


9.595 006 


0.012 455 


0.024 088 


0.417 


0.036 713 


9.632 650 


9.595 937 


0.012 514 


0.024 199 


0.418 


0.036 885 


9.633 750 


9.596 866 


0.012 574 


0.024 311 


0.419 


0.037 057 


9.634 848 


9.597 791 


0.012 634 


0.024 423 


0.420 


0.037 229 


9.635 943 


9.598 714 


0.012 694 


0.024 535 


0.421 


0.037 401 


9.637 036 


9.599 635 


0.012 754 


0.024 647 


0.422 


0.037 574 


9.638 127 


9.600 553 


0.012 814 


0.024 760 


0.423 


0.037 748 


9.639 215 


9.601 468 


0.012 875 


0.024 873 


0.424 


0.037 921 


9.640 301 


9.602 380 


0.012 936 


0.024 986 


0.425 


0.038 095 


9.641 385 


9.603 290 


0.012 996 


0.025 099 


0.426 


0.038 270 


9.642 467 


9.604 197 


0.013 057 


0.025 213 


0.427 


0.038 445 


9.643 546 


9.605 101 


0.013 118 


0.025 326 


0.428 


0.038 620 


9.644 623 


9.606 003 


0.013 179 


0.025 440 


0.429 


0.038 795 


9.645 698 


9.606 903 


0.013 241 


0.025 555 


0.430 


0.038 971 


9.646 770 


9.607 799 


0.013 302 


0.025 669 


0.431 


0.039 147 


9.647 841 


9.608 694 


0.013 364 


0.025 784 


0.432 


0.039 324 


9.648 909 


9.609 585 


0.013 425 


0.025 899 


0.433 


0.039 501 


9.649 975 


9.610 474 


0.013 487 


0.026 014 


0.434 


0.039 678 


9.651039 


9.611 361 


0.013 549 


0.026 129 


0.435 


0.039 856 


9.652 100 


9.612 245 


0.013 611 


0.026 244 


0.436 


0.040 034 


9.653 160 


9.613 126 


0.013 673 


0.026 360 


0.437 


0.040 212 


9.654 217 


9.614 005 


0.013 736 


0.026 476 


0.438 


0.040 391 


9.655 273 


9.614 882 


0.013 798 


0.026 592 


0.439 


0.040 570 


9.656 326 


9.615 756 


0.013 861 


0.026 709 


0.440 


0.040 749 


9.657 377 


9.616 627 


0.013 924 


0.026 825 


0.441 


0.040 929 


9.658 425 


9.617 496 


0.013 987 


0.026 942 


0.442 


0.041 109 


9.659 472 


9.618 363 


0.014 050 


0.027 059 


0.443 


0.041 290 


9.660 517 


9.619 227 


0.014 113 


0.027 177 


0.444 


0.041 471 


9.661 560 


9.620 089 


0.014 177 


0.027 294 


0.445 


0.041 652 


9.662 600 


9.620 948 


0.014 240 


0.027 412 


0.446 


0.041 834 


9.663 639 


9.621805 


0.014 304 


0.027 530 


0.447 


0.042 016 


9.664 675 


9.622 659 


0.014 368 


0.027 648 


0.448 


0.042 198 


9.665 709 


9.623 511 


0.014 431 


0.027 767 


0.449 


0.042 381 


9.666 742 


9.624 361 


0.014 496 


0.027 885 



294 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



X 


log cosh z 


log Hlllh 1 


log tanh x 


, .-mil X 

log 

* 


tanh x 


0.450 


0.042 564 


9.667 772 


9.625 209 


0.014 560 


0.028 004 


0.451 


0.042 747 


9.668 801 


9.626 053 


0.014 624 


0.028 123 


0.452 


0.042 931 


9.669 827 


9.626 896 


0.014 689 


0.028 242 


0.453 


0.043 115 


9.670 851 


9.627 736 


0.014 753 


0.028 362 


0.454 


0.043 300 


9.671 874 


9.628 574 


0.014 818 


0.028 482 


0.455 


0.043 484 


9.672 894 


9.629 410 


0.014 883 


0.028 601 


0.456 


0.043 670 


9.673 913 


9.630 243 


0.014 948 


0.028 722 


0.457 


0.043 855 


9.674 929 


9.631074 


0.015 013 


0.028 842 


0.458 


0.044 041 


9.675 944 


9.631 903 


0.015 078 


0.028 963 


0.459 


0.044 227 


9.676 957 


9.632 729 


0.015 144 


0.029 083 


0.460 


0.044 414 


9.677 967 


9.633 553 


0.015 210 


0.029 204 


0.461 


0.044 601 


9.678 976 


9.634 375 


0.015 275 


0.029 326 


0.462 


0.044 788 


9.679 983 


9.635 195 


0.015 341 


0.029 447 


0.463 


0.044 976 


9.680 988 


9.636 012 


0.015 407 


0.029 569 


0.464 


0.045 164 


9.681991 


9.636 827 


0.015 473 


0.029 690 


0.465 


0.045 352 


9.682 993 


9.637 640 


0.015 540 


0.029 813 


0.466 


0.045 541 


9.683 992 


9.638 451 


0.015 606 


0.029 935 


0.467 


0.045 730 


9.684 989 


9.639 260 


0.015 673 


0.030 057 


0.468 


0.045 919 


9.685 985 


9.640 066 


0.015 739 


0.030 180 


0.469 


0.046 109 


9.686 979 


9.640 870 


0.015 806 


0.030 303 


0.470 


0.046 299 


9.687 971 


9.641672 


0.015 873 


0.030 426 


0.471 


0.046 490 


9.688 961 


9.642 471 


0.015 940 


0.030 550 


0.472 


0.046 681 


9.689 950 


9.643 269 


0.016 008 


0.030 673 


0.473 


0.046 872 


9.690 936 


9.644 064 


0.016 075 


0.030 797 


0.474 


0.047 063 


9.691 921 


9.644 857 


0.016 143 


0.030 921 


0.475 


0.047 255 


9.692 904 


9.645 649 


0.016 210 


0.031045 


0.476 


0.047 447 


9.693 885 


9.646 437 


0.016 278 


0.031 169 


0.477 


0.047 640 


9.694 864 


9.647 224 


0.016 346 


0.031294 


0.478 


0.047 833 


9.695 842 


9.648 009 


0.016 414 


0.031419 


0.479 


0.048 026 


9.696 818 


9.648 792 


0.016 482 


0.031544 


0.480 


0.048 220 


9.697 792 


9.649 572 


0.016 551 


0.031669 


0.481 


0.048 414 


9.698 764 


9.650 350 


0.016 619 


0.031795 


0.482 


0.048 608 


9.699 735 


9.651 127 


0.016 688 


0.031920 


0.483 


0.048 803 


9.700 704 


9.651901 


0.016 757 


0.032 046 


0.484 


0.048 998 


9.701671 


9.652 673 


0.016 826 


0.032 172 


0.485 


0.049 193 


9.702 636 


9.653 443 


0.016 895 


0.032 299 


0.486 


0.049 389 


9.703 600 


9.654 211 


0.016 964 


0.032 425 


0.487 


0.049 585 


9.704 562 


9.654 977 


0.017 033 


0.032 552 


0.488 


0.049 781 


9.705 522 


9.655 741 


0.017 103 


0.032 679 


0.489 


0.049 978 


9.706 481 


9.656 503 


0.017 172 


0.032 806 


0.490 


0.050 175 


9.707 438 


9.657 263 


0.017 242 


0.032 933 


0.491 


0.050 373 


9.708 393 


9.658 021 


0.017 312 


0.033 061 


0.492 


0.050 570 


9.709 347 


9.658 777 


0.017 382 


0.033 189 


0.493 


0.050 769 


9.710 299 


9.659 530 


0.017 452 


0.033 316 


0.494 


0.050 967 


9.711 249 


9.660 282 


0.017 522 


0.033 445 


0.495 


0.051 166 


9.712 198 


9.661 032 


0.017 593 


0.033 573 


0.496 


0.051 365 


9.713 145 


9.661 780 


0.017 663 


0.033 702 


0.497 


0.051 565 


9.714 091 


9.662 526 


0.017 734 


0.033 830 


0.498 


0.051 764 


9.715 034 


9.663 270 


0.017 805 


0.033 959 


0.499 


0.051 965 


9.715 977 


9.664 012 


0.017 876 


0.034 089 



APPENDIX 



295 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log si li li X 



log tanh x 



0.500 
0.501 
0.502 
0.503 
0.504 
0.505 
0.506 
0.507 
0.508 
0.509 
0.510 
0.511 
0.512 
0.513 
0.514 
0.515 
0.516 
0.517 
0.518 
0.519 
0.520 
0.521 
0.522 
0.523 
0.524 
0.525 
0.526 
0.527 
0.528 
0.529 
0.530 
0.531 
0.532 
0.533 
0.534 
0.535 
0.536 
0.537 
0.538 
0.539 

0.540 
0.541 
0.542 
0.543 
0.544 
0.545 
0.546 
0.547 
0.548 
0.549 



0.052 165 
0.052 366 
0052 567 
0.052 769 
0.052 971 
0.053 173 
0.053 375 
0.053 578 
0.053 782 
0.053 985 
0.054 189 
0.054 393 
0.054 598 
0.054 803 
0.055 008 
.055 214 
0.055 420 
0.055 626 
0.055 833 
0.056 040 
0.056 247 
0.056 454 
0.056 662 
0.056 871 
0.057 079 
0.057 288 
0.057 498 
0.057 707 
0.057 917 
0.058 128 
0.058338 
0.058 549 
0.058 760 
0.058 972 
0.059 184 
0.059 396 
0.059 609 
0.059 822 
0.060 "035 
0.060 249 
0.060 463 
0.060 677 
0.060 892 
0.061 106 
0.061322 
0.061 537 
0.061 753 
0.061 969 
0.062 186 
0.062 403 



201 
201 
202 
202 
202 
202 
203 
204 
203 
204 
204 
205 
205 
205 
206 
206 
206 
207 
207 
207 
207 
208 
209 
208 
209 
210 
209 
210 
211 
210 
211 
211 
212 
212 
212 
213 
213 
213 
214 
214 
214 
215 
214 
216 
215 
216 
216 
217 
217 
217 



9.716 917 

9.717 856 

9.718 794 

9.719 729 

9.720 664 

9.721 596 

9.722 527 

9.723 457 

9.724 385 

9.725 311 

9.726 236 

9.727 160 

9.728 081 

9.729 002 

9.729 921 

9.730 838 

9.731 754 

9.732 668 

9.733 581 

9.734 492 

9.735 402 

9.736 311 

9.737 217 

9.738 123 

9.739 027 

9.739 930 

9.740 831 

9.741 730 

9.742 629 

9.743 526 

9.744 421 

9.745 315 

9.746 208 

9.747 099 

9.747 989 

9.748 877 

9.749 764 

9.750 650 

9.751 534 

9.752 417 

9.753 299 

9.754 179 

9.755 058 

9.755 936 

9.756 812 

9.757 687 

9.758 561 

9.759 433 

9.760 304 

9.761 173 



939 

938 
935 
935 
932 
931 
930 
928 
926 
925 
924 
921 
921 
919 
917 
916 
914 
913 
911 
910 
909 
906 
906 
904 
903 
901 
899 
899 
897 
895 
894 
893 
891 
890 
888 
887 
886 
884 
883 
882 

880 
879 
878 
876 
875 
874 
872 
871 
869 
869 



9.664 752 

9.665 490 

9.666 226 

9.666 961 

9.667 693 

9.668 423 

9.669 152 

9.669 879 

9.670 603 
9.671326 
9.672 047 

9.672 766 

9.673 484 

9.674 199 

9.674 913 

9.675 624 

9.676 334 

9.677 042 

9.677 748 

9.678 453 

9.679 155 

9.679 856 

9.680 555 

9.681 252 

9.681 948 

9.682 641 

9.683 333 

9.684 023 

9.684 711 

9.685 398 

9.686 083 

9.686 766 

9.687 447 

9.688 127 

9.688 805 

9.689 481 

9.690 155 

9.690 828 

9.691 499 

9.692 169 

9.692 836 

9.693 502 

9.694 167 

9.694 829 

9.695 490 

9.696 150 

9.696 807 

9.697 463 

9.698 118 
9.698 770 



738 
736 
735 
732 
730 
729 
727 
724 
723 
721 
719 
718 
715 
714 
711 
710 
708 
706 
705 
702 
701 
699 
697 
696 
693 
692 
690 
688 
687 
685 
683 
681 
680 
678 
676 
674 
673 
671 
670 
667 
666 
665 
662 
661 
660 
657 
656 
655 
652 
652 



290 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 
(Continued) 



log coah z 



J 



log sinh x 



log tanh x 



0.550 
0.551 
0.552 
0.553 
0.554 
0.555 
0.556 
0.557 
0.558 
0.559 
0.560 
0.561 
0.562 
0.563 
0.564 
0.565 
0.566 
0.567 
0.568 
0.569 
0.570 
0.571 
0.572 
0.573 
0.574 
0.575 
0.576 
0.577 
0.578 
0.579 
0.580 
0.581 
0.582 
0.583 
0.584 
0.585 
0.586 
0.587 
0.588 
0.589 
0.590 
0.591 
0.592 
0.593 
0.594 
0.595 
0.596 
0.597 
0.598 
0.599 



0.062 620 
0.062 838 
0.063 056 
0.063 274 
0.063 492 
0.063 711 
0.063 930 
0.064 150 
0.064 369 
0.064 590 
0.064 810 
0.065 031 
0.065 252 
0.065 473 
0.065 695 
0.065 917 
0.066 140 
0.066 362 
0.066 585 
0.066 809 
0.067 032 
0.067 256 
0.067 481 
0.067 705 
0.067 930 
0.068 155 
0.068 381 
0.068 607 
0.068 833 
0.069 059 
0.069 286 
0.069 513 
0.069 741 
0.069 969 
0.070 197 
0.070 425 
0.070 654 
0.070 883 
0.071 112 
0.071 342 
0.071 572 
0.071 802 
0.072 033 
0.072 264 
0.072 495 
0.072 726 
0.072 958 
0.073 190 
0.073 423 
0.073 656 



218 
218 
218 
218 
219 
219 
220 
219 
221 
220 
221 
221 
221 
222 
222 
223 
222 
223 
224 
223 
224 
225 
224 
225 
225 
226 
226 
226 
226 
227 
227 
228 
228 
228 
228 
229 
229 
229 
230 
230 
230 
231 
231 
231 
231 
232 
232 
233 
233 
233 



9.762 042 

9.762 909 

9.763 775 

9.764 639 

9.765 502 

9.766 364 

9.767 225 

9.768 084 

9.768 942 

9.769 799 

9.770 654 

9.771 509 

9.772 362 

9.773 214 

9.774 064 

9.774 914 

9.775 762 

9.776 609 

9.777 455 

9.778 299 

9.779 142 

9.779 984 

9.780 825 

9.781 665 

9.782 504 

9.783 341 

9.784 177 

9.785 012 

9.785 846 

9.786 679 

9.787 510 

9.788 340 

9.789 170 

9.789 998 

9.790 825 

9.791 650 

9.792 475 

9.793 298 

9.794 121 

9.794 942 

9.795 762 

9.796 581 

9.797 399 

9.798 216 

9.799 032 

9.799 846 

9.800 660 

9.801 472 

9.802 284 

9.803 094 



867 
866 
864 
863 
862 
861 
859 
858 
857 
855 
855 
853 
852 
850 
850 
848 
847 
846 
844 
843 
842 
841 
840 
839 
837 
836 
835 
834 
833 
831 
830 
830 
828 
827 
825 
825 
823 
823 
821 
820 
819 
818 
817 
816 
814 
814 
812 
812 
810 
809 



9.699 422 

9.700 071 

9.700 719 

9.701 365 

9.702 010 

9.702 653 

9.703 294 

9.703 934 

9.704 573 

9.705 209 

9.705 844 

9.706 478 

9.707 110 

9.707 740 

9.708 369 

9.708 997 

9.709 622 

9.710 247 

9.710 869 

9.711 490 

9.712 110 

9.712 728 

9.713 345 

9.713 960 

9.714 574 

9.715 186 

9.715 796 

9.716 405 

9.717 013 

9.717 619 

9.718 224 

9.718 827 

9.719 429 

9.720 029 

9.720 628 

9.721 225 

9.721 821 

9.722 416 

9.723 009 

9.723 600 

9.724 190 

9.724 779 

9.725 366 

9.725 952 

9.726 537 

9.727 120 

9.727 702 

9.728 282 

9.728 861 

9.729 438 



649 
648 
646 
645 
643 
641 
640 
639 
636 
635 
634 
632 
630 
629 
628 
625 
625 
622 
621 
620 
618 
617 
615 
614 
612 
610 
609 
608 
606 
605 
603 
602 
600 
599 
597 
596 
595 
593 
591 
590 
589 
587 
586 
585 
583 
582 
580 
579 
577 
576 



APPENDIX 



297 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh x 



0.600 
0.601 
0.602 
0.603 
0.604 
0.605 
0.606 
0.607 
0.608 
0.609 
0.610 
0.611 
0.612 
0.613 
0.614 
0.615 
0.616 
0.617 
0.618 
0.619 
0.620 
0.621 
0.622 
0.623 
0.624 
0.625 
0.626 
0.627 
0.628 
0.629 
0.630 
0.631 
0.632 
0.633 
0.634 
0.635 
0.636 
0.637 
0.638 
0.639 
0.640 
0.641 
0.642 
0.643 
0.644 
0.645 
0.646 
0.647 
0.648 
0.649 



0.073 889 
0.074 122 
0.074 356 
0.074 590 
0.074 824 
0.075 059 
0.075 294 
0.075 529 
0.075 765 
0.076 000 

0.076 237 
0.076 473 
0.076 710 
0.076 947 
0.077 184 
0.077 422 
0.077 660 
0.077 898 
0.078 137 
0.078 376 
0.078 615 
0.078 854 
0.079 094 
0.079 334 
0.079 575 
0.079 815 
0.080 056 
0.080 298 
0.080 539 
0.080 781 
0.081024 
0.081 266 
0.081509 
0.081 752 
0.081 995 
0.082 239 
0.082 483 
0.082 727 
0.082 972 
0.083 217 
0.083 462 
0.083 707 
0.083 953 
0.084 199 
0.084 446 
0.084 692 
0.084 939 
0.085 187 
0.085 434 
0.085 682 



233 
234 
234 
234 
235 
235 
235 
236 
235 
237 
236 
237 
237 
237 
238 
238 
238 
239 
239 
239 
239 
240 
240 
241 
240 
241 
242 
241 
242 
243 
242 
243 
243 
243 
244 
244 
244 
245 
245 
245 
245 
246 
246 
247 
246 
247 
248 
247 
248 
248 



9.803 903 

9.804 711 

9.805 518 

9.806 324 

9.807 129 

9.807 933 

9.808 736 

9.809 538 

9.810 339 

9.811 138 

9.811 937 

9.812 735 

9.813 531 

9.814 327 

9.815 121 

9.815 915 

9.816 707 

9.817 499 

9.818 289 

9.819 079 

9.819 867 

9.820 655 
9.821441 

9.822 227 

9.823 012 

9.823 795 

9.824 578 

9.825 359 

9.826 140 

9.826 920 

9.827 698 

9.828 476 

9.829 253 

9.830 029 

9.830 804 

9.831 578 

9.832 351 

9.833 123 

9.833 894 

9.834 664 

9.835 433 

9.836 202 

9.836 969 

9.837 735 

9.838 501 

9.839 266 

9.840 029 

9.840 792 

9.841 554 

9.842 315 



808 
807 
806 
805 
804 
803 
802 
801 
799 
799 
798 
796 
796 
794 
794 
792 
792 
790 
790 
788 
788 
786 
786 
785 
783 
783 
781 
781 
780 
778 
778 
777 
776 
775 
774 
773 
772 
771 
770 
769 
769 
767 
766 
766 
765 
763 
763 
762 
761 
760 



9.730 014 

9.730 589 

9.731 162 
9.731734 

9.732 305 

9.732 874 

9.733 442 

9.734 009 

9.734 574 

9.735 138 

9.735 700 

9.736 262 

9.736 821 

9.737 380 

9.737 937 

9.738 493 

9.739 047 

9.739 601 

9.740 153 

9.740 703 

9.741 252 

9.741 801 

9.742 347 

9.742 893 

9.743 437 

9.743 980 

9.744 521 

9.745 062 

9.745 601 

9.746 138 

9.746 675 

9.747 210 

9.747 744 

9.748 277 

9.748 808 

9.749 339 

9.749 868 

9.750 395 
9.750 922 
9.751447 
9.751971 

9.752 494 
9.753016 

9.753 536 

9.754 055 

9.754 573 

9.755 090 

9.755 606 

9.756 120 
9.756 633 



575 
573 
572 
571 
569 
568 
567 
565 
564 
562 
562 
559 
559 
557 
556 
554 
554 
552 
550 
549 
549 
546 
546 
544 
543 
541 
541 
539 
537 
537 
535 
534 
533 
531 
531 
529 
527 
527 
525 
524 
523 
522 
520 
519 
518 
517 
516 
514 
513 
512 



298 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



1(>K COMll I 



log sinh z 



log tanh z 



0.650 


0.085 930 


0.651 


0.086 178 


0.652 


0.086 427 


0.653 


0.086 676 


0.654 


0.086 925 


0.655 


0.087 175 


0.656 


0.087 425 


0.657 


0.087 675 


0.658 


0.087 926 


0.659 


0.088 176 


0.660 


0.088 427 


0.661 


0.088 679 


0.662 


0.088 930 


0.663 


0.089 182 


0.664 


0.089 434 


0.665 


0.089 687 


0.666 


0.089 940 


0.667 


0.090 193 


0.668 


0.090 446 


0.669 


0.090 700 


0.670 


0.090 954 


0.671 


0.091 208 


0.672 


0.091462 


0.673 


0.091717 


0.674 


0.091 972 


0.675 


0.092 227 


0.676 


0.092 483 


0.677 


0.092 739 


0.678 


0.092 995 


0.679 


0.093 252 


0.680 


0.093 508 


0.681 


0.093 765 


0.682 


0.094 023 


0.683 


0.094 280 


0.684 


0.094 538 


0.685 


0.094 796 


0.686 


0.095 055 


0.687 


0.095 313 


0.688 


0.095 572 


0.689 


0.095 832 


0.690 


. 0.096 091 


0.691 


0.096 351 


0.692 


0.096 611 


0.693 


0.096 872 


0.694 


0.097 132 
0.097 393 


0.695 


0.696 


0.097 655 


0.697 


0.097 916 


0.698 


0.098 178 


0.699 


0.098 440 



248 
249 
249 
249 
250 
250 
250 
251 
250 
251 
252 
251 
252 
252 
253 
253 
253 
253 
254 
254 
254 
254 
255 
255 
255 
256 
256 
256 
257 
256 
257 
258 
257 
258 
258 
259 
258 
259 
260 
259 
260 
260 
261 
260 
261 
262 
261 
262 
262 
262 



9.843 075 

9.843 835 

9.844 593 

9.845 351 

9.846 107 

9.846 863 

9.847 618 

9.848 372 

9.849 125 

9.849 877 

9.850 628 

9.851 379 

9.852 128 

9.852 877 

9.853 625 

9.854 372 

9.855 118 

9.855 863 

9.856 608 

9.857 352 

9.858 094 

9.858 836 

9.859 578 

9.860 318 

9.861 057 

9.861 796 

9.862 534 

9.863 271 

9.864 007 

9.864 743 

9.865 477 

9.866 211 

9.866 944 

9.867 676 

9.868 408 

9.869 138 

9.869 868 

9.870 597 

9.871 325 

9.872 053 

9.872 779 

9.873 505 

9.874 230 

9.874 955 

9.875 678 

9.876 401 

9.877 123 

9.877 844 

9.878 565 

9.879 285 



760 
758 
758 
756 
756 
755 
754 
753 
752 
751 
751 
749 
749 
748 
747 
746 
745 
745 
744 
742 
742 
742 
740 
739 
739 
738 
737 
736 
736 
734 
734 
733 
732 
732 
730 
730 
729 
728 
728 
726 
726 
725 
725 
723 
723 
722 
721 
721 
720 
719 



9.757 145 

9.757 656 

9.758 166 

9.758 674 

9.759 182 

9.759 688 

9.760 193 

9.760 696 

9.761 199 

9.761 701 

9.762 201 

9.762 700 

9.763 198 

9.763 695 

9.764 191 

9.764 685 

9.765 179 

9.765 671 

9.766 162 

9.766 652 

9.767 141 

9.767 629 

9.768 115 

9.768 601 

9.769 085 

9.769 569 

9.770 051 

9.770 532 
9.771012 
9.771491 

9.771 969 

9.772 446 

9.772 921 

9.773 396 

9.773 870 

9.774 342 

9.774 813 

9.775 284 

9.775 753 

9.776 221 

9.776 688 

9.777 154 

9.777 619 

9.778 083 

9.778 546 

9.779 008 
9.779 469 

9.779 928 

9.780 387 
9.780 845 



511 
510 
508 
508 
506 
505 
503 
503 
502 
500 
499 
498 
497 
496 
494 
494 
492 
491 
490 
489 
488 
486 
486 
484 
484 
482 
481 
480 
479 
478 
477 
475 
475 
474 
472 
471 
471 
469 
468 
467 
466 
465 
464 
463 
462 
461 
459 
459 
458 
456 



APPENDIX 



299 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh x 



0.700 
0.701 
0.702 
0.703 
0.704 
0.705 
0.706 
0.707 
0.708 
0.709 
0.710 
0.711 
0.712 
0.713 
0.714 
0.715 
0.716 
0.717 
0.718 
0.719 
0.720 
0.721 
0.722 
0.723 
0.724 
0.725 
0.726 
0.727 
0.728 
0.729 
0.730 
0.731 
0.732 
0.733 
0.734 
0.735 
0.736 
0.737 
0.738 
0.739 
0.740 
0.741 
0.742 
0.743 
0.744 
0.745 
0.746 
0.747 
0.748 
0.749 



0.098 702 
0.098 965 
0.099 228 
0.099 491 
0.099 754 
0.100 018 
0.100 282 
0.100 546 
0.100 811 
0.101076 
0.101341 
0.101606 
0.101872 
0.102 138 
0.102 404 
0.102 670 
0.102 937 
0.103 204 
0.103 471 
0.103 739 
0.104 006 
0.104 274 
0.104 543 
0.104 811 
0.105 080 
0.105 349 
0.105 619 
0.105 888 
0.106 158 
0.106 429 
0.106 699 
0.106 970 
0.107 241 
0.107 512 
0.107 783 
0.108 055 
0.108 327 
0.108 600 
0.108 872 
0.109 145 
0.109 418 
0.109 691 
0.109 965 
0.110 239 
0.110 513 
0.110 788 
0.111062 
0.111337 
0.111612 
0.111888 



263 
263 
263 
263 
264 
264 
264 
265 
265 
265 
265 
266 
266 
266 
266 
267 
267 
267 
268 
267 
268 
269 
268 
269 
269 
270 
269 
270 
271 
270 
271 
271 
271 
271 
272 
272 
273 
272 
273 
273 
273 
274 
274 
274 
275 
274 
275 
275 
276 
276 



9.880 004 

9.880 722 

9.881 439 

9.882 156 

9.882 872 

9.883 587 

9.884 302 

9.885 015 

9.885 728 

9.886 441 

9.887 152 

9.887 863 

9.888 573 

9.889 282 

9.889 991 

9.890 699 

9.891 406 

9.892 113 

9.892 818 

9.893 523 

9.894 228 

9.894 931 

9.895 634 

9.896 336 

9.897 038 

9.897 739 

9.898 439 

9.899 138 

9.899 837 

9.900 535 
9.901233 

9.901 929 

9.902 625 

9.903 321 

9.904 015 

9.904 709 

9.905 402 

9.906 095 

9.906 787 

9.907 478 

9.908 169 

9.908 859 

9.909 548 

9.910 237 

9.910 925 

9.911 612 

9.912 299 

9.912 985 

9.913 670 

9.914 355 



718 
717 
717 
716 
715 
715 
713 
713 
713 
711 
711 
710 
709 
709 
708 
707 
707 
705 
705 
705 
703 
703 
702 
702 
701 
700 
699 
699 
698 
698 
696 
696 
696 
694 
694 
693 
693 
692 
691 
691 
690 
689 
689 
688 
687 
687 
686 
685 
685 
684 



9.781 301 

9.781 757 

9.782 211 

9.782 665 

9.783 118 

9.783 569 

9.784 020 
9.784 469 

9.784 918 

9.785 365 

9.785 811 

9.786 257 

9.786 701 

9.787 145 

9.787 587 

9.788 029 
9.788 469 

9.788 909 

9.789 347 

9.789 785 

9.790 221 

9.790 657 

9.791 091 

9.791 525 
9.791958 

9.792 389 

9.792 820 

9.793 250 

9.793 679 

9.794 107 
9.794 534 

9.794 960 

9.795 385 

9.795 809 

9.796 232 

9.796 654 

9.797 075 
9.797 496 

9.797 915 

9.798 333 

9.798 751 

9.799 167 
9.799 583 

9.799 998 

9.800 412 

9.800 825 

9.801 237 
9.801648 

9.802 058 
9.802 467 



456 
454 
454 
453 
451 
451 
449 
449 
447 
446 
446 
444 
444 
442 
442 
440 
440 
438 
438 
436 
436 
434 
434 
433 
431 
431 
430 
429 
428 
427 
426 
425 
424 
423 
422 
421 
421 
419 
418 
418 
416 
416 
415 
414 
413 
412 
411 
410 
409 
409 



300 



AIMMADIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh i 



i 



log sinh z 



log tanh i 



i 



0.750 
0.751 
0.752 
0.753 
0.754 
0.755 
0.756 
0.757 
0.758 
0.759 
0.760 
0.761 
0.762 
0.763 
0.764 
0.765 
0.766 
0.767 
0.768 
0.769 
0.770 
0.771 
0.772 
0.773 
0.774 
0.775 
0.776 
0.777 
0.778 
0.779 
0.780 
0.781 
.782 
.783 
.784 
.785 
.786 
0.787 
0.788 
0.789 
0.790 
0.791 
0.792 
0.793 
0.794 
0.795 
0.796 
0.797 
0.798 
0.799 



0.112 164 
0.112 440 
0.112 716 
0.112 992 
0.113 269 
0.113 546 
0.113 823 
0.114 101 
0.114 379 
0.114 657 
0.114 935 
0.115 213 
0.115 492 
0.115 771 
0.116 051 
0.116 330 
0.116 610 
0.116 890 
0.117 170 
0.117 451 
0.117 732 
0.118 013 
0.118 294 
0.118 576 
0.118 858 
0.119 140 
0.119 422 
0.119 705 
0.119 987 
0.120 271 
0.120 554 
0.120 837 
0.121 121 
0.121 405 
0.121690 
0.121974 
0.122 259 
0.122 544 
0.122 830 
0.123 115 

0.123 401 
0.123 687 
0.123 973 
0.124 260 
0.124 547 
0.124 834 
0.125 121 
0.125 409 
0.125 696 
0.125 984 



276 
276 
276 
277 

277 
277 
278 
278 
278 
278 
278 
279 
279 
280 
279 
280 
280 
280 
281 
281 
281 
281 
282 
282 
282 
282 
283 
282 
284 
283 
283 
284 
284 
285 
284 
285 
285 
286 
285 
286 

286 
286 
287 
287 
287 
287 
288 
287 
288 
289 



9.915 039 

9.915 723 

9.916 405 

9.917 088 

9.917 769 

9.918 450 

9.919 130 

9.919 810 

9.920 489 

9.921 167 

9.921 845 

9.922 522 

9.923 199 

9.923 875 

9.924 550 

9.925 225 

9.925 899 

9.926 572 

9.927 245 

9.927 917 

9.928 589 

9.929 260 

9.929 930 

9.930 600 
9.931269 
9.931938 

9.932 606 

9.933 273 

9.933 940 

9.934 606 

9.935 272 

9.935 937 

9.936 602 

9.937 265 

9.937 929 

9.938 592 

9.939 254 

9.939 915 

9.940 576 

9.941 237 

9.941897 

9.942 556 

9.943 215 

9.943 873 

9.944 531 

9.945 188 

9.945 844 

9.946 500 

9.947 156 
9.947 810 



684 
682 
683 
681 
681 
680 
680 
679 
678 
678 
677 
677 
676 
675 
675 
674 
673 
673 
672 
672 
671 
670 
670 
669 
669 
668 
667 
667 
666 
666 
665 
665 
663 
664 
663 
662 
661 
661 
661 
660 

659 
659 
658 
658 
657 
656 
656 
656 
654 
655 



9.802 876 

9.803 283 

9.803 690 

9.804 095 
9.804 500 

9.804 904 

9.805 307 

9.805 709 

9.806 110 
9.806 511 

9.806 910 

9.807 309 

9.807 706 

9.808 103 
9.808 499 

9.808 894 

9.809 289 

9.809 682 

9.810 074 
9.810 466 

9.810 857 

9.811 247 
9.811636 

9.812 024 
9.812 412 

9.812 798 

9.813 184 
9.813 569 

9.813 953 

9.814 336 

9.814 718 

9.815 100 
9.815 480 

9.815 860 

9.816 239 
9.816 617 

9.816 995 

9.817 371 

9.817 747 

9.818 122 

9.818 496 

9.818 869 

9.819 241 
9.819 613 

9.819 984 

9.820 354 

9.820 723 

9.821 092 
9.821 459 
9.821 826 



407 
407 
405 
405 
404 
403 
402 
401 
401 
399 
399 
397 
397 
396 
395 
395 
393 
392 
392 
391 
390 
389 
388 
388 
386 
386 
385 
384 
383 
382 
382 
380 
380 
379 
378 
378 
376 
376 
375 
374 

373 
372 
372 
371 
370 
369 
369 
367 
367 
366 



APPENDIX 



301 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh x 



0.800 
0.801 
0.802 
0.803 
0.804 
0.805 
0.806 
0.807 
0.808 
0.809 
0.810 
0.811 
0.812 
0.813 
0.814 
0.815 
0.816 
0.817 
0.818 
0.819 
0.820 
0.821 
0.822 
0.823 
0.824 
0.825 
0.826 
0.827 
0.828 
0.829 
0.830 
0.831 
0.832 
0.833 
0.834 
0.835 
0.836 
0.837 
0.838 
0.839 
0.840 
0.841 
0.842 
0.843 
0.844 
0.845 
0.846 
0.847 
0.848 
0.849 



0.126 273 
0.126 561 
0.126 850 
0.127 139 
0.127 428 
0.127 718 
0.128 007 
0.128 297 
0.128 588 
0.128 878 
0.129 169 
0.129 460 
0.129 751 
0.130 042 
0.130 334 
0.130 626 
0.130 918 
0.131210 
0.131503 
0.131795 
0.132 089 
0.132 382 
0.132 675 
0.132 969 
0.133 263 
0.133 557 
0.133 852 
0.134 147 
0.134 442 
0.134 737 
0.135 032 
0.135 328 
0.135 624 
0.135 920 
0.136 216 
0.136 513 
0.136 809 
0.137 106 
0.137 404 
0.137 701 
0.137 999 
0.138 297 
0.138 595 
0.138 894 
0.139 192 
0.139 491 
0.139 790 
0.140 090 
0.140 389 
0.140 689 



288 
289 
289 
289 
290 
289 
290 
291 
290 
291 
291 
291 
291 
292 
292 
292 
292 
293 
292 
294 
293 
293 
294 
294 
294 
295 
295 
295 
295 
295 
296 
296 
296 
296 
297 
296 
297 
298 
297 
298 

298 
298 
299 
298 
299 
299 
300 
299 
300 
300 



9.948 465 

9.949 119 

9.949 772 

9.950 424 
9.951076 

9.951 728 

9.952 379 

9.953 030 

9.953 679 

9.954 329 

9.954 978 

9.955 626 

9.956 274 

9.956 921 

9.957 568 

9.958 214 

9.958 860 

9.959 505 

9.960 150 

9.960 794 

9.961 437 

9.962 080 

9.962 723 

9.963 365 

9.964 006 

9.964 647 

9.965 288 

9.965 928 

9.966 567 

9.967 206 

9.967 845 

9.968 483 

9.969 120 

9.969 757 

9.970 394 
9.971030 

9.971 665 

9.972 300 

9.972 935 

9.973 569 

9.974 202 

9.974 835 

9.975 468 

9.976 100 
9.976-731 

9.977 362 

9.977 993 

9.978 623 

9.979 253 
9.979 882 



654 
653 
652 
652 
652 
651 
651 
649 
650 
649 
648 
648 
647 
647 
646 
646 
645 
645 
644 
643 
643 
643 
642 
641 
641 
641 
640 
639 
639 
639 
638 
637 
637 
637 
636 
635 
635 
635 
634 
633 

633 
633 
632 
631 
631 
631 
630 
630 
629 
629 



9.822 192 
9.822 557 

9.822 922 

9.823 285 

9.823 648 

9.824 010 
9.824 372 

9.824 732 

9.825 092 
9.825 451 

9.825 809 

9.826 167 
9.826 523 

9.826 879 

9.827 234 
9.827 588 

9.827 942 

9.828 295 
9.828 647 

9.828 998 

9.829 349 

9.829 699 

9.830 048 
9.830 396 

9.830 743 
9.831090 

9.831 436 

9.831 781 

9.832 126 
9.832 470 

9.832 813 

9.833 155 
9.833 497 

9.833 838 

9.834 178 
9.834 517 

9.834 856 

9.835 194 
9.835 531 

9.835 867 

9.836 203 
9.836 538 

9.836 873 

9.837 206 
9.837 539 

9.837 871 

9.838 203 
9.838 534 

9.838 864 

9.839 193 



365 
365 
363 
363 
362 
362 
360 
360 
359 
358 
358 
356 
356 
355 
354 
354 
353 
352 
351 
351 
350 
349 
348 
347 
347 
346 
345 
345 
344 
343 
342 
342 
341 
340 
339 
339 
338 
337 
336 
336 
335 
335 
333 
333 
332 
332 
331 
330 
329 
329 



302 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



0.850 
0.851 
0.852 
0.853 
0.854 
0.855 
0.856 
0.857 
0.858 
0.859 
0.860 
0.861 
0.862 
0.863 
0.864 
0.865 
0.S66 
0.867 
0.868 
0.869 
0.870 
0.871 
0.872 
0.873 
0.874 
0.875 
0.876 
0.877 
0.878 
0.879 
0.880 
0.881 
0.882 
0.883 
0.884 
0.885 
0.886 
0.887 
0.888 
0.889 
0.890 
0.891 
0.892 
0.893 
0.894 
0.895 
0.896 
0.897 
0.898 
0.899 



log cosh i 



0.140 989 
0.141289 
0.141590 
0.141890 
0.142 191 
0.142 492 
0.142 794 
0.143 095 
0.143 397 
0.143 699 
0.144 001 
0.144 304 
0.144 607 
0.144 910 
0.145 213 
0.145 516 
0.145 820 
0.146 124 
0.146 428 
0.146 732 

0.147 036 
0.147 341 
0.147 646 
0.147 951 
0.148 257 
0.148 562 
0.148 868 
0.149 174 
0.149 480 
0.149 787 
0.150 093 
0.150 400 
0.150 707 
0.151015 
0.151322 
0.151 630 
0.151 938 
0.152 246 
0.152 555 
0.152 863 
0.153 172 
0.153 481 
0.153 790 
0.154 100 
0.154 410 
0.154 720 
0.155 030 
0.155 340 
0.155 651 
0.155 961 



300 
301 
300 
301 
301 
302 
301 
302 
302 
302 
303 
303 
303 
303 
303 
304 
304 
304 
304 
304 
305 
305 
305 
306 
305 
306 
306 
306 
307 
306 
307 
307 
308 
307 
308 
308 
308 
309 
308 
309 
309 
309 
310 
310 
310 
310 
310 
311 
310 
311 



log sinh x 



9.980 511 

9.981 139 

9.981 766 

9.982 394 

9.983 021 

9.983 647 

9.984 273 

9.984 898 

9.985 523 

9.986 147 

9.986 771 

9.987 395 

9.988 018 

9.988 641 

9.989 263 

9.989 884 

9.990 506 

9.991 126 

9.991 747 

9.992 367 

9.992 986 

9.993 605 

9.994 224 

9.994 842 

9.995 459 

9.996 077 

9.996 693 

9.997 310 

9.997 926 

9.998 541 

9.999 156 
9.999 771 
0.000 385 
0.000 998 
0.001 612 
0.002 224 
0.002 837 
0.003 449 
0.004 060 
0.004 671 
0.005 282 
0.005 892 
0.006 502 
0.007 112 
Q.007 721 
0.008 329 
0.008 938 
0.009 545 
0.010 153 
0.010 760 



628 
627 
628 
627 
626 
626 
625 
625 
624 
624 
624 
623 
623 
622 
621 
622 
620 
621 
620 
619 
619 
619 
618 
617 
618 
616 
617 
616 
615 
615 
615 
614 
613 
614 
612 
613 
612 
611 
611 
611 
610 
610 
610 
609 
608 
609 
607 
608 
607 
606 



log t.-illll z 



9.839 522 

9.839 850 

9.840 177 
9.840 503 

9.840 829 

9.841 155 
9.841 479 

9.841 803 

9.842 126 
9.842 448 

9.842 770 

9.843 091 
9.843 411 

9.843 731 

9.844 050 
9.844 368 

9.844 686 

9.845 003 
9.845 319 
9.845 635 

9.845 950 

9.846 264 
9.846 578 

9.846 891 

9.847 203 
9.847 514 

9.847 825 

9.848 136 
9.848 445 

9.848 754 

9.849 063 
9.849 370 
9.849 677 

9.849 984 

9.850 289 
9.850 594 

9.850 899 

9.851 203 
9.851506 

9.851 808 

9.852 110 
9.852 411 

9.852 712 

9.853 012 
9.853 311 
9.853 610 

9.853 908 

9.854 205 
9.854 502 
9.854 798 



328 
327 
326 
326 
326 
324 
324 
323 
322 
322 
321 
320 
320 
319 
318 
318 
317 
316 
316 
315 
314 
314 
313 
312 
311 
311 
311 
309 
309 
309 
307 
307 
307 
305 
305 
305 
304 
303 
302 
302 
301 
301 
300 
299 
299 
298 
297 
297 
296 
296 



APPENDIX 



303 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh x 



0.900 
0.901 
0.902 
0.903 
0.904 
0.905 
0.906 
0.907 
0.908 
0.909 
0.910 
0.911 
0.912 
0.913 
0.914 
0.915 
0.916 
0.917 
0.918 
0.919 
0.920 
0.921 
0.922 
0.923 
0.924 
0.925 
0.926 
0.927 
0.928 
0.929 
0.930 
0.931 
0.932 
0.933 
0.934 
0.935 
0.936 
0.937 
0.938 
0.939 
0.940 
0.941 
0.942 
0.943 
0.944 
0.945 
0.946 
0.947 
0.948 
0.949 



0.156 272 
0.156 584 
0.156 895 
0.157 207 
0.157 518 
0.157 830 
0.158 143 
0.158 455 
0.158 768 
0.159 081 
0.159 394 
0.159 707 
0.160 021 
0.160 334 
0.160 648 
0.160 962 
0.161277 
0.161591 
0.161906 
0.162 221 
0.162 536 
0.162 851 
0.163 167 
0.163 483 
0.163 799 
0.164 115 
0.164 431 
0.164 748 
0.165 065 
0.165 382 
0.165 699 
0.166 016 
0.166 334 
0.166 651 
0.166 969 
0.167 288 
0.167 606 
0.167 925 
0.168 243 
0.168 563 
0.168 882 
0.169 201 
0.169 521 
0.169 841 
0.170 161 
0.170 481 
0.170 801 
0.171 122 
0.171442 
0.171763 



312 
311 
312 
311 
312 
313 
312 
313 
313 
313 
313 
314 
313 
314 
314 
315 
314 
315 
315 
315 
315 
316 
316 
316 
316 
316 
317 
317' 
317 
317 
317 
318 
317 
318 
319 
318 
319 
318 
320 
319 
319 
320 
320 
320 
320 
320 
321 
320 
321 
322 



0.011 366 
0.011 972 
0.012 578 
0.013 183 
0.013 788 
0.014 392 
0.014 996 
0.015 600 
0.016 203 
0.016 806 
0.017 409 
0.018 011 
0.018 612 
0.019 214 
0.019 814 
0.020 415 
0.021 015 
0.021 615 
0.022 214 
0.022 813 
0.023 411 
0.024 009 
0.024 607 
0.025 204 
0.025 801 
0.026 398 
0.026 994 
0.027 590 
0.028 185 
0.028 780 
0.029 375 
0.029 969 
0.030 563 
0.031 156 
0.031 749 
0.032 342 
0.032 935 
0.033 527 
0.034 118 
0.034 709 
0.035 300 
0.035 891 
0.036 481 
0.037 071 
0.037 660 
0.038 249 
0.038 838 
0.039 426 
0.040 014 
0.040 602 



606 
606 
605 
605 
604 
604 
604 
603 
603 
603 
602 
601 
602 
600 
601 
600 
600 
599 
599 
598 
598 
598 
597 
597 
597 
596 
596 
595 
595 
595 
594 
594 
593 
593 
593 
593 
592 
591 
591 
591 
591 
590 
590 
589 
589 
589 
588 
588 
588 
587 



9.855 094 
9.855 389 
9.855 683 

9.855 977 

9.856 270 
9.856 562 

9.856 854 

9.857 145 
9.857 436 

9.857 726 

9.858 015 
9.858 304 
9.858 592 

9.858 879 

9.859 166 
9.859 453 

9.859 738 

9.860 023 
9.860 308 
9.860 592 

9.860 875 

9.861 158 
9.861 440 

9.861 722 

9.862 003 
9.862 283 
9.862 563 

9.862 842 

9.863 121 
9.863 399 

9.863 676 

9.863 953 

9.864 229 
9.864 505 

9.864 780 

9.865 055 
9.865 328 
9.865 602 

9.865 875 

9.866 147 
9.866 419 
9.866 690 

9.866 960 

9.867 230 
9.867 500 

9.867 769 

9.868 037 
9.868 305 
9.868 572 
9.868 838 



295 
294 
294 
293 
292 
292 
291 
291 
290 
289 
289 
288 
287 
287 
287 
285 
285 
285 
284 
283 
283 
282 
282 
281 
280 
280 
279 
279 
278 
277 
277 
276 
276 
275 
275 
273 
274 
273 
272 
272 
271 
270 
270 
270 
269 
268 
268 
267 
266 
266 



304 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



leg nak i 



log sinh z 



log tanh z 



0.960 

0.951 

0.952 

0.953 

0.954 

0.955 

0.956 

0.957 

0.958 

0.959 

0.960 

0.961 

0.962 

0.963 

0.964 

0.965 

0.966 

0.967 

0.968 

0.969 

0.970 

0.971 

0.972 

0.973 

0.974 

0.975 

0.976 

0.977 

0.978 

0.979 

0.980 

0.981 

0.982 

0.983 

0.984 

0.985 





0.988 

0.989 

0.990 

0.991 

0.992 

0.993 

0.994 

0.995 

0.996 

0.997 

0.998 

0.999 



987 



0.172 085 
0.172 406 
0.172 728 
0.173 049 
0.173 371 
0.173 694 
0.174 016 
0.174 338 
0.174 661 
0.174 984 
0.175 307 
0.175 631 
0.175 954 
0.176 278 
0.176 602 
0.176 926 
0.177 250 
0.177 575 
0.177 899 
0.178 224 

0.178 549 
0.178 875 
0.179 200 
0.179 526 
0.179 851 
0.180 177 
0.180 504 
0.180 830 
0.181 157 
0.181483 
0.181810 
0.182 137 
0.182 465 
0.182 792 
0.183 120 
0.183 448 
0.183 776 
0.184 104 
0.184 433 
0.184 761 
0.185 090 
0.185 419 
0.185 748 
0.186 078 
0.186 407 
0.186 737 
0.187 067 
0.187 397 
0.187 727 
0.188 058 



321 
322 
321 
322 
323 
322 
322 
323 
323 
323 
324 
323 
324 
324 
324 
324 
325 
324 
325 
325 
326 
325 
326 
325 
326 
327 
326 
327 
326 
327 
327 
328 
327 
328 
328 
328 
328 
329 
328 
329 
329 
329 
330 
329 
330 
330 
330 
330 
331 
331 



0.041 189 
0.041 776 
0.042 362 
0.042 949 
0.043 534 
0.044 120 
0.044 705 
0.045 290 
0.045 874 
0.046 458 
0.047 042 
0.047 625 
0.048 208 
0.048 791 
0.049 373 
0.049 955 
0.050 537 
0.051 118 
0.051 699 
0.052 279 
0.052 860 
0.053 439 
0.054 019 
0.054 598 
0.055 177 
0.055 756 
0.056 334 
0.056 912 
0.057 489 
0.058 066 
0.058 643 
0.059 220 
0.059 796 
0.060 372 
0.060 947 
0.061 523 
0.062 098 
0.062 672 
0.063 246 
0.063 820 
0.064 394 
0.064 967 
0.065 540 
0.066 113 
0.066 685 
0.067 257 
0.067 829 
0.068 400 
0.068 971 
0.069 542 



587 
5N6 
587 
585 
586 
585 
585 
584 
584 
584 
583 
583 
583 
582 
582 
582 
581 
581 
580 
581 
579 
580 
579 
579 
579 
578 
578 
577 
577 
577 
577 
576 
576 
575 
576 
575 
574 
574 
574 
574 
573 
573 
573 
572 
572 
572 
571 
571 
571 
570 



9.869 104 
9.869 370 
9.869 635 

9.869 899 

9.870 163 
9.870 426 
9.870 689 

9.870 951 

9.871 213 
9.871 474 

9.871 735 
9.871995 

9.872 254 
9.872 513 

9.872 771 

9.873 029 
9.873 286 
9.873 543 

9.873 799 

9.874 055 
9.874 310 
9.874 565 

9.874 819 

9.875 073 
9.875 326 
9.875 578 

9.875 830 

9.876 082 
9.876 333 
9.876 583 

9.876 833 

9.877 082 
9.877 331 
9.877 580 

9.877 827 

9.878 075 
9.878 322 
9.878 568 

9.878 814 

9.879 059 
9.879 304 
9.879 548 

9.879 792 

9.880 035 
9.880 278 
9.880 520 

9.880 762 

9.881 003 
9.881 244 
9.881484 



266 
265 
264 
264 
263 
263 
262 
262 
261 
261 
260 
259 
259 
258 
258 
257 
257 
256 
256 
255 
255 
254 
254 
253 
252 
252 
252 
251 
250 
250 
249 
249 
249 
247 
248 
247 
246 
246 
245 
245 
244 
244 
243 
243 
242 
242 
241 
241 
240 
240 



APPENDIX 



305 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh i 



log sinh x 



log tanh x 



1.000 
1.001 
1.002 
1.003 
1.004 
1.005 
1.006 
1.007 
1.008 
1.009 
1.010 
1.011 
1.012 
1.013 
1.014 
1.015 
1.016 
1.017 
1.018 
1.019 
1.020 
1.021 
1.022 
1.023 
1.024 
1.025 
1.026 
1.027 
1.028 
1.029 
1.030 
1.031 
1.032 
1.033 
1.034 
1.035 
1.036 
1.037 
1.038 
1.039 
1.040 
1.041 
1.042 
1.043 
1.044 
1.045 
1.046 
1.047 
1.048 
1.049 



0.188 389 
0.188 719 
0.189 050 
0.189 382 
0.189 713 
0.190 045 
0.190 376 
0.190 708 
0.191040 
0.191373 
0.191705 
0.192 038 
0.192 371 
0.192 704 
0.193 037 
0.193 370 
0.193 704 
0.194 038 
0.194 372 
0.194 706 
0.195 040 
0.195 374 
0.195 709 
0.196 044 
0.196 379 
0.196 714 
0.197 049 
0.197 385 
0.197 720 
0.198 056 
0.198 392 
0.198 728 
0.199 065 
0.199 401 
0.199 738 
0.200 075 
0.200 412 
0.200 749 
0.201 087 
0.201 424 
0.201 762 
0.202 100 
0.202 438 
0.202 776 
0.203 115 
0.203 453 
0.203 792 
0.204 131 
0.204 470 
0.204 809 



330 
331 
332 
331 
332 
331 
332 
332 
333 
332 
333 
333 
333 
333 
333 
334 
334 
334 
334 
334 
334 
335 
335 
335 
335 
335 
336 
335 
336 
336 
336 
337 
336 
337 
337 
337 
337 
338 
337 
338 
338 
338 
338 
339 
"338 
339 
339 
339 
339 
340 



0.070 112 
0.070 682 
0.071 252 
0.071 822 
0.072 391 
0.072 960 
0.073 528 
0.074 096 
0.074 664 
0.075 232 
0.075 799 
0.076 366 
0.076 933 
0.077 499 
0.078 065 
0.078 631 
0.079 196 
0.079 762 
0.080 326 
0.080 891 
0.081455 
0.082 019 
0.082 583 
0.083 146 
0.083 709 
0.084 272 
0.084 835 
0.085 397 
0.085 959 
0.086 520 
0.087 082 
0.087 643 
0.088 203 
0.088 764 
0.089 324 
0.089 884 
0.090 443 
0.091 003 
0.091 562 
0.092 121 
0.092 679 
0.093 237 
0.093 795 
0.094 353 
0.094 910 
0.095 467 
0.096 024 
0.096 580 
0.097 136 
0.097 692 



570 
570 
570 
569 
569 
568 
568 
568 
568 
567 
567 
567 
566 
566 
566 
565 
566 
564 
565 
564 
564 
564 
563 
563 
563 
563 
562 
562 
561 
562 

561 
560 
561 
560 
560 
559 
560 
559 
559 
558 
558 
558 
558 
557 
557 
557 
556 
556 
556 
556 



9.881724 
9.881963 
-9.882 202 
9.882 440 
9.882 678 

9.882 915 

9.883 152 
9.883 388 
9.883 624 

9.883 859 

9.884 094 
9.884 328 
9.884 562 

9.884 795 

9.885 028 
9.885 261 
9.885 493 
9.885 724 

9.885 955 

9.886 185 
9.886 415 
9.886 645 

9.886 874 

9.887 103 
9.887 331 
9.887 558 

9.887 786 

9.888 012 
9.888 238 
9.888 464 
9.888 690 

9.888 914 

9.889 139 
9.889 363 
9.889 586 

9.889 809 

9.890 032 
9.890 254 
9.890 475 
9.890 696 

9.890 917 

9.891 137 
9.891357 

9.891 576 
9.891795 

9.892 014 
9.892 232 
9.892 449 
9.892 666 
9.892 883 



239 
239 
238 
238 
237 
237 
236 
236 
235 
235 
234 
234 
233 
233 
233 
232 
231 
.231 
230 
230 
230 
229 
229 
228 
227 
228 
226 
226 
226 
226 
224 
225 
224 
223 
223 
223 
222 
221 
221 
221 
220 
220 
219 
219 
219 
218 
217 
217 
217 
216 



306 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh z 



1.050 
1.051 
1.052 
1.053 
1.054 
1.055 
1.056 
1.057 
1.058 
1.059 
1.060 
1.061 
1.062 
1.063 
1.064 
1.065 
1.066 
1.067 
1.068 
1.069 
1.070 
1.071 
1.072 
1.073 
1.074 
1.075 
1.076 
1.077 
1.078 
1.079 
1.080 
1.081 
1.082 
1.083 
1.084 
1.085 
1.086 
1.087 
1.088 
1.089 
1.090 
1.091 
1.092 
1.093 
1.094 
1.095 
1.096 
1.097 
1.098 
1.099 



0.205 149 
0.205 488 
0.205 828 
0.206 168 
0.206 508 
0.206 848 
0.'207 189 
0.207 530 
0.207 870 
0.208 211 
0.208 552 
0.208 894 
0.209 235 
0.209 577 
0.209 919 
0.210 261 
0.210 603 
0.210 945 
0.211 287 
0.211 630 
0.211 973 
0.212 316 
0.212 659 
0.213 002 
0.213 346 
0.213 689 
0.214 033 
0.214 377 
0.214 721 
0.215 065 
0.215 410 
0.215 754 
0.216 099 
0.216 444 
0.216 789 
0.217 134 
0.217 479 
0.217 825 
0.218 171 
0.218 516 
0.218 862 
0.219 209 
0.219 555 
0.219 901 
0.220 248 
0.220 595 
0.220 942 
0.221 289 
0.221 636 
0.221 983 



339 
340 
340 
340 
340 
341 
341 
340 
341 
341 
342 
341 
342 
342 
342 
342 
342 
342 
343 
343 
343 
343 
343 
344 
343 
344 
344 
344 
344 
345 
344 
345 
345 
345 
345 
345 
346 
346 
345 
346 
347 
346 
346 
347 
347 
347 
347 
347 
347 
348 



0.098 248 
0.098 803 
0.099 358 
0.099 913 
0.100 468 
0.101 022 
0.101576 
0.102 130 
0.102 683 
0.103 236 
0.103 789 
0.104 342 
0.104 894 
0.105 446 
0.105 998 
0.106 550 
0.107 101 
0.107 652 
0.108 203 
0.108 753 
0.109 304 
0.109 854 
0.110 403 
0.110 953 
0.111502 
0.112 051 
0.112 600 
0.113 148 
0.113 696 
0.114 244 
0.114 792 
0.115 339 
0.115 886 
0.116 433 
0.116 980 
0.117 526 
0.118 072 
0.118 618 
0.119 164 
0.119 709 
0.120 254 
0.120 799 
0.121344 
0.121888 
0.122 432 
0.122 976 
0.123 520 
0.124 063 
0.124 606 
0.125 149 



555 
555 
555 
555 
554 
554 
554 
553 
553 
553 
553 
552 
552 
552 
552 
551 
551 
551 
550 
551 
550 
549 
550 
549 
549 
549 
548 
548 
548 
548 
547 
547 
547 
547 
546 
546 
546 
546 
545 
545 
545 
545 
544 
544 
544 
544 
543 
543 
543 
543 



9.893 099 
9.893 315 
9.893 530 
9.893 745 

9.893 960 

9.894 173 
9.894 387 
9.894 600 

9.894 813 

9.895 025 
9.895 237 
9.895 448 
9.895 659 

9.895 870 

9.896 080 
9.896 289 
9.896 498 
9.896 707 

9.896 915 

9.897 123 
9.897 331 
9.897 538 
9.897 744 

9.897 951 

9.898 156 
9.898 362 
9.898 567 
9.898 771 

9.898 975 

9.899 179 
9.899 382 
9.899 585 
9.899 787 

9.899 989 

9.900 191 
9.900 392 
9.900 593 
9.900 793 

9.900 993 

9.901 193 
9.901 392 
9.901 591 
9.901 789 

9.901 987 

9.902 184 
9.902 381 
9.902 578 
9.902 774 

9.902 970 

9.903 166 



216 
215 
215 
215 
213 
214 
213 
213 
212 
212 
211 
211 
211 
210 
209 
209 
209 
208 
208 
208 
207 
206 
207 
205 
206 
205 
204 
204 
204 
203 
203 
202 
202 
202 
201 
201 
200 
200 
200 
199 
199 
198 
198 
197 
197 
197 
196 
196 
196 
195 



APPENDIX 



307 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



1.100 
1.101 
1.102 
1.103 
1.104 
1.105 
1.106 
1.107 
1.108 
1.109 
1.110 
1.111 
1.112 



113 
114 
115 
116 



1.117 
1.118 
1.119 
1.120 
1.121 
1.122 
1.123 
1.124 
1.125 
1.126 
1.127 
1.128 
1.129 
1.130 
1.131 
1.132 
1.133 
1.134 
1.135 
1.136 
1.137 
1.138 
1.139 
1.140 
1.141 
1.142 
1.143 
1.144 



145 
146 
147 
148 
149 



log cosh i 



0.222 331 
0.222 679 
0.223 027 
0.223 375 
0.223 723 
0.224 071 
0.224 420 
0.224 768 
0.225 117 
0.225 466 
0.225 815 
0.226 165 
0.226 514 
0.226 864 
0.227 213 
0.227 563 
0.227 913 
0.228 263 
0.228 614 
0.228 964 
0.229 315 
0.229 666 
0.230 017 
0.230 368 
0.230 719 
0.231070 
0.231422 
0.231 774 
0.232 126 
0.232 478 
0.232 830 
0.233 182 
0.233 534 
0.233 887 
0.234 240 
0.234 593 
0.234 946 
0.235 299 
0.235 652 
0.236 006 
0.236 359 
0.236 713 
0.237 067 
0.237 421 
0.237 775 
0.238130 
0.238 484 
0.238 839 
0.239 194 
0.239 548 



348 
348 
348 
348 
348 
349 
348 
349 
349 
349 
350 
349 
350 
349 
350 
350 
350 
351 
350 
351 
351 
351 
351 
351 
351 
352 
352 
352 
352 
352 
352 
352 
353 
353 
353 
353 
353 
353 
354 
353 
354 
354 
354 
354 
355 
354 
355 
355 
354 
356 



log sinh x 



0.125 692 
0.126 234 
0.126 776 
0.127 318 
0.127 860 
0.128 401 
0.128 943 
0.129 484 
0.130 024 
0.130 565 
0.131 105 
0.131645 
0.132 185 
0.132 724 
0.133 264 
0.133 803 
0.134 342 
0.134 880 
0.135 419 
0.135 957 
0.136 495 
0.137 032 
0.137 570 
0.138 107 
0.138 644 
0.139 181 
0.139 717 
0.140 253 
0.140 789 
0.141325 
0.141861 
0.142 396 
0.142 931 
0.143 466 
0.144 001 
0.144 536 
0.145 070 
0.145 604 
0.146 138 
0.146 671 
0.147 205 
0.147 738 
0.148 271 
0.148 803 
0.149 336 
0.149 868 
0.150 400 
0.150 932 
0.151464 
0.151995 



542 
542 
542 
542 
541 
542 
541 
540 
541 
540 
540 
540 
539 
540 
539 
539 
538 
539 
538 
538 
537 
538 
537 
537 
537 
536 
536 
536 
536 
536 
535 
535 
535 
535 
535 
534 
534 
534 
533 
534 
533 
533 
532 
533 
532 
532 
532 
532 
531 
531 



log tanh x 



9.903 361 
9.903 555 
9.903 750 

9.903 944 

9.904 137 
9.904 330 
9.904 523 
9.904 715 

9.904 907 

9.905 099 
9.905 290 
9.905 480 
9.905 671 

9.905 861 

9.906 050 
9.906 240 

9.905 428 

9.906 617 
9.906 805 

9.906 992 

9.907 180 
9.907 366 
9.907 553 
9.907 739 

9.907 925 

9.908 110 
9.908 295 
9.908 480 
9.908 664 

9.908 848 

9.909 031 
9.909 214 
9.909 397 
9.909 579 
9.909 761 

9.909 943 

9.910 124 
9.910 305 
9.910 485 
9.910 666 

9.910 845 
9.911025 

9.911 204 
9.911 382 
9.911561 

9.911 739 
9.911916 

9.912 093 
9.912 270 
9.912 447 



194 
195 
194 
193 
193 
193 
192 
192 
192 
191 
190 
191 
190 
189 
190 
188 
189 
188 
187 
188 
186 
187 
186 
186 
185 
185 
185 
184 
184 
183 
183 
183 
182 
182 
182 
181 
181 
180 
181 
179 
180 
179 
178 
179 
178 
177 
177 
177 
177 
176 



:*os 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 
(Continued) 



1.150 
1.151 
1.152 
1.153 
1.154 



1.155 
1.156 
1.157 
1.158 
1.159 
1.160 
1.161 
1.162 
1.163 
1.164 
1.165 
1.166 
1.167 
1.168 
1.169 
1.170 
1.171 
1.172 
1.173 
1.174 
1.175 
1.176 
1.177 
1.178 
1.179 
1.180 
1.181 
1.182 
1.183 
1.184 
1.185 
1.186 
1.187 
1.188 
1.189 



.190 

.191 

.192 

.193 

.194 

.195 

1.196 

1.197 

1.198 

1.199 



log cosh z 



0.239 904 
0.240 259 
0.240 614 
0.240 970 
0.241325 
0.241 681 
0.242 037 
0.242 393 
0.242 749 
0.243 106 
0.243 462 
0.243 819 
0.244 176 
0.244 532 
0.244 890 
0.245 247 
0.245 604 
0.245 962 
0.246 319 
0.246 677 
0.247 035 
0.247 393 
0.247 751 
0.248109 
0.248 468 
0.248 827 
0.249 185 
0.249 544 
0.249 903 
0.250 262 
0.250 622 
0.250 981 
0.251341 
0.251 700 
0.252 060 
0.252 420 
0.252 780 
0.253 140 
0.253 501 
0.253 861 
0.254 222 
0.254 583 
0.254 944 
0.255 305 
0.255 666 
0.256 027 
0.256 389 
0.256 750 
0.257112 
0.257 474 



355 
355 
356 
355 
356 
356 
356 
356 
357 
356 
357 
357 
356 
358 
357 
357 
358 
357 
358 
358 
358 
358 
358 
359 
359 
358 
359 
359 
359 
360 
359 
360 
359 
360 
360 
360 
360 
361 
360 
361 
361 
361 
361 
361 
361 
362 
361 
362 
362 
362 



log sinh z 



0.152 526 
0.153 057 
0.153 588 
0.154 119 
0.154 649 
0.155 179 
0.155 709 
0.156 239 
0.156 768 
0.157 297 
0.157 826 
0.158 355 
0.158 884 
0.159 412 
0.159 941 
0.160 469 
0.160 996 
0.161524 
0.162 051 
0.162 579 
0.163 106 
0.163 632 
0.164 159 
0.164 685 
0.165 211 
0.165 737 
0.166 263 
0.166 789 
0.167 314 
0.167 839 
0.168 364 
0.168 889 
0.169 414 
0.169 938 
0.170 462 
0.170 986 
0.171510 
0.172 033 
0.172 557 
0.173 080 
0.173 603 
0.174 126 
0.174 648 
0.175 171 
0.175 693 
0.176 215 
0.176 737 
0.177 258 
0.177 780 
0.178 301 



531 
531 
531 
530 
530 
530 
530 
529 
529 
529 
529 
529 
528 
529 
528 
527 
528 
527 
528 
527 
526 
527 
526 
526 
526 
526 
526 
525 
525 
525 
525 
525 
524 
524 
524 
524 
523 
524 
523 
523 
523 
522 
523 
522 
522 
522 
521 
522 
521 
521 



log tanh x 



9.912 623 
9.912 798 

9.912 974 

9.913 149 
9.913 324 
9.913 498 
9.913 672 

9.913 846 

9.914 019 
9.914 192 
9.914 364 
9.914 536 
9.914 708 

9.914 880 

9.915 051 
9.915 222 
9.915 392 
9.915 562 
9.915 732 

9.915 902 

9.916 071 
9.916 239 
9.916 408 
9.916 576 
9.916 744 

9.916 911 

9.917 078 
9.917 245 
9.917 411 
9.917 577 
9.917 743 

9.917 908 

9.918 073 
9.918 238 
9.918 402 
9.918 566 
9.918 730 

9.918 893 

9.919 056 
9.919 219 
9.919 381 
9.919 543 
9.919 705 

9.919 866 

9.920 027 
9.920 188 
9.920 348 
9.920 508 
9.920 668 
9.920 827 



7li 
75 
75 
74 
71 
71 
73 
73 
72 
72 
72 
72 
71 
71 
70 
70 
70 
70 
00 
08 
00 
os 

OS 
07 
07 

07 

00 
00 
00 
05 
05 
05 
04 
04 
01 
03 

03 

03 
02 
02 
02 
01 
(11 
01 
00 
00 

00 

50 
59 



APPENDIX 



309 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh z 



1.200 
1.201 
1.202 
1.203 
1.204 
1.205 
1.206 
1.207 
1.208 
1.209 



.210 
.211 
.212 
.213 
.214 
.215 
1.216 
1.217 
1.218 
1.219 
1.220 
1.221 
1.222 
1.223 
1.224 
1.225 
1.226 
1.227 
1.228 
1.229 
1.230 
1.231 
1.232 
1.233 
1.234 
1.235 
1.236 
1.237 
1.238 
1.239 
1.240 
1.241 
1.242 
1.243 
1.244 
1.245 
1.246 
1.247 
1.248 
1.249 



0.257 836 
0.258 198 
0.258 560 
0.258 923 
0.259 285 
0.259 648 
0.260 011 
0.260 373 
0.260 736 
0.261 100 
0.261463 
0.261 826 
0.262 190 
0.262 554 
0.262 917 
0.263 281 
0.263 645 
0.264 010 
0.264 374 
0.264 738 
0.265 103 
0.265 468 
0.265 833 
0.266 198 
0.266 563 
0.266 928 
0.267 293 
0.267 659 
0.268 024 
0.268 390 
0.268 756 
0.269 122 
0.269 488 
0.269 854 
0.270 221 
0.270 587 
0.270 954 
0.271 321 
0.271 687 
0.272 054 
0.272 422 
0.272 789 
0.273 156 
0.273 524 
0.273 891 
0.274 259 
0.274 627 
0.274 995 
0.275 363 
0.275 731 



362 
362 
363 
362 
363 
363 
362 
363 
364 
363 
363 
364 
364 
363 
364 
364 
365 
364 
364 
365 
365 
365 
365 
365 
365 
365 
366 
365 
366 
366 
366 
366 
366 
367 
366 
367 
367 
366 
367 
368 
367 
367 
368 
367 
368 
368 
368 
368 
368 
368 



0.178 822 
0.179 343 
0.179 864 
0.180 384 
0.180 904 
0.181424 
0.181944 
0.182 464 
0.182 984 
0.183 503 
0.184 022 
0.184 541 
0.185 060 
0.1S5 578 
0.186 097 
0.186 615 
0.187 133 
0.187 651 
0.188 169 
0.188 686 
0.189 204 
0.189 721 
0.190 238 
0.190 754 
0.191271 
0.191787 
0.192 304 
0.192 820 
0.193 336 
0.193 851 
0.194 367 
0.194 882 
0.195 397 
0.195 912 
0.196 427 
0.196 942 
0.197 456 
0.197 971 
0.198 485 
0.198 999 
0.199 512 
0.200 026 
0.200 539 
0.201053 
0.201 566 
0.202 079 
0.202 591 
0.203 104 
0.203 616 
0.204129 



521 
521 
520 
520 
520 
520 
520 
520 
519 
519 
519 
519 
518 
519 
518 
518 
518 
518 
517 
518 
517 
517 
516 
517 
516 
517 
516 
516 
515 
516 
515 
515 
515 
515 
515 
514 
515 
514 
514 
513 
514 
513 
514 
513 
513 
512 
513 
512 
513 
512 



9.920 986 

9.921 145 
9.921 303 
9.921 461 
9.921 619 
9.921 777 

9.921 934 

9.922 091 
9.922 247 
9.922 403 
9.922 559 
9.922 715 

9.922 870 

9.923 025 
9.923 179 
9.923 334 
9.923 488 
9.923 641 
9.923 795 

9.923 948 

9.924 100 
9.924 253 
9.924 405 
9.924 557 
9.924 708 

9.924 859 

9.925 010 
9.925 161 
9.925 311 
9.925 461 
9.925 611 
9.925 760 

9.925 909 

9.926 058 
9.926 206 
9.926 355 
9.926 502 
9.926 650 
9.926 797 

9.926 944 

9.927 091 
9.927 237 
9.927 383 
9.927 529 
9.927 674 
9.927 820 

9.927 965 

9.928 109 
9.928 253 
9.928 397 



59 
.58 
58 
58 
58 
57 
57 
56 
56 
56 
56 
55 
55 
54 
55 
54 
53 
54 
53 
52 
53 
52 
52 
51 
51 
51 
51 
50 
50 
50 
49 
49 
49 
48 
49 
47 
48 
47 
47 
47 
46 
46 
46 
45 
48 
45 
44 
44 
44 
44 



310 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

{Continued) 



!<>>• cosh i 



log sinh z 



log tanh z 



1.250 

1.251 

1.252 

1.253 

1.254 

1.255 

1^56 

1.257 

1.258 

1.259 

1.260 

1.261 

1.262 

1.263 

1.264 

1.265 

1.266 

1.267 

1.268 

1.269 

1.270 

1.271 

1.272 

1.273 

1.274 

1.275 

1.276 

1.277 

1.278 

1.279 

1.280 

1.281 

1.282 

1.283 

1.284 

1.285 

1.286 

1.287 

1.288 

1.289 

1.290 

1.291 

1.292 

1.293 

1.294 

1.295 

1.296 

1.297 

1.298 

1.299 



0.276 099 
0.276 468 
0.276 837 
0.277 205 
0.277 574 
0.277 943 
0.278 312 
0.278 681 
0.279 051 
0.279 420 
0.279 790 
0.280 159 
0.280 529 
0.280 899 
0.281 269 
0.281 639 
0.282 009 
0.282 380 
0.282 750 
0.283 121 
0.283 492 
0.283 863 
0.284 233 
0.284 605 
0.284 976 
0.285 347 
0.285 719 
0.286 090 
0.286 462 
0.286 834 
0.287 206 
0.287 578 
0.287 950 
0.288 322 
0.288 694 
0.289 067 
0.289 439 
0.289 812 
0.290 185 
0.290 558 
0.290 931 
0.291304 
0.291 677 
0.292 051 
0.292 424 
0.292 798 
0.293 172 
0.293 546 
0.293 919 
0.294 294 



369 
369 
368 
369 
369 
369 
369 
370 
369 
370 
369 
370 
370 
370 
370 
370 
371 
370 
371 
371 
371 
370 
372 
371 
371 
372 
371 
372 
372 
372 
372 
372 
372 
372 
373 
372 
373 
373 
373 
373 
373 
373 
374 
373 
374 
374 
374 
373 
375 
374 



0.204 641 
0.205 152 
0.205 664 
0.206 176 
0.206 687 
0.207 198 
0.207 709 
0.208 220 
0.208 731 
0.209 241 
0.209 752 
0.210 262 
0.210 772 
0.211 282 
0.211 792 
0.212 301 
0.212 811 
0.213 320 
0.213 829 
0.214 338 
0.214 847 
0.215 355 
0.215864 
0.216 372 
0.216 880 
0.217 388 
0.217 896 
0.218 403 
0.218 911 
0.219 418 
0.219 925 
0.220 432 
0.220 939 
0.221 446 
0.221 952 
0.222 459 
0.222 965 
0.223 471 
0.223 977 
0.224 482 
0.224 988 
0.225 493 
0.225 999 
0.226 504 
0.227 009 
0.227 514 
0.228 018 
0.228 523 
0.229 027 
0.229 531 



511 
512 
512 
511 
511 
511 
511 
511 
510 
511 
510 
510 
510 
510 
509 
510 
509 
509 
509 
509 
508 
509 
508 
508 
508 
508 
507 
508 
507 
507 
507 
507 
507 
506 
507 
506 
506 
506 
505 
506 
505 
506 
505 
505 
505 
504 
505 
504 
504 
504 



9.928 541 
9.928 685 
9.928 828 

9.928 970 

9.929 113 
9.929 255 
9.929 397 
9.929 539 
9.929 680 
9.929 821 

9.929 962 

9.930 103 
9.930 243 
9.930 383 
9.930 523 
9.930 662 
9.930 801 

9.930 940 

9.931 079 
9.931217 
9.931 355 
9.931 493 
9.931 630 
9.931 767 

9.931 904 

9.932 041 
9.932 177 
9.932 313 
9.932 449 
9.932 584 
9.932 720 
9.932 855 

9.932 989 

9.933 124 
9.933 258 
9.933 392 
9.933 525 
9.933 659 
9.933 792 

9.933 925 

9.934 057 
9.934 189 
9.934 321 
9.934 453 
9.934 585 
9.934 716 
9.934 847 

9.934 977 

9.935 108 
9.935 238 



11 
43 
42 
43 
42 
42 
42 
41 
41 
41 
41 
40 
40 
40 
39 
39 
39 
39 
88 
38 
88 
87 
37 
37 
87 
86 
36 
36 
35 
36 

36 

34 
36 
34 

34 
33 
34 
83 
33 
32 
32 
■■',2 
32 
32 
31 
31 
30 
31 
30 
30 



APPENDIX 



311 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

, (Continued) 



log cosh x 



log sinh x 



log tanh i 



1.300 

1.301 

1.302 

1.303 

1.304 

1.305 

1.306 

1.307 

1.308 

1.309 

1.310 

1.311 

1.312 

1.313 

1.314 

1.315 

1.316 

1.317 

1.318 

1.319 

1.320 

1.321 

1.322 

1.323 

1.324 

1.325 

1.326 

1.327 

1.328 

1.329 

1.330 

1.331 

1.332 

1.333 

1.334 

1.335 

1.336 

1.337 

1.338 

1.339 

1.340 

1.341 

1.342 

1.343 

1.344 

1.345 

1.346 

1.347 

1.348 

1.349 



0.294 668 
0.295 042 
0.295 416 
0.295 791 
0.296 166 
0.296 540 
0.296 915 
0.297 290 
0.297 665 
0.298 040 
0.298 416 
0.298 791 
0.299 167 
0.299 542 
0.299 918 
0.300 294 
0.300 670 
0.301046 
0.301422 
0.301798 
0.302 175 
0.302 551 
0.302 928 
0.303 304 
0.303 681 
0.304 058 
0.304 435 
0.304 812 
0.305 190 
0.305 567 
0.305 944 
0.306 322 
0.306 700 
0.307 077 
0.307 455 
0.307 833 
0.308 211 
0.308 590 
0.308 968 
0.309 346 
0.309 725 
0.310 103 
0.310 482 
0.310 861 
0.311 240 
0.311 619 
0.311 998 
0.312 377 
0.312 757 
0.313 136 



374 

374 

375 

375 

374 

375 

375 

375 

375 

376 

375 

376 

375 

376 

376 

376 

376 

376 

376 

377 

376 

377 

376 

377 

377 

377 

377 

378 

377 

377 

378 

378 

377 

378 

378 

378 

379 

378 

378 

379 

378 

379 

379 

379 

379 

379 

379 

380 

379 

380 



0.230 035 
0.230 539 
0.231 043 
0.231 547 
0.232 050 
0.232 554 
0.233 057 
0.233 560 
0.234 063 
0.234 565 
0.235 068 
0.235 570 
0.236 073 
0.236 575 
0.237 077 
0.237 579 
0.238 080 
0.238 582 
0.239 083 
0.239 584 
0.240 086 
0.240 586 
0.241 087 
0.241 588 
0.242 089 
0.242 589 
0.243 089 
0.243 589 
0.244 089 
0.244 589 
0.245 089 
0.245 588 
0.246 088 
0.246 587 
0.247 086 
0.247 585 
0.248 084 
0.248 583 
0.249 081 
0.249 580 



0.250 
0.250 
0.251 
0.251 
0.252 
0.252 
0.253 
0.253 
0.254 
0.254 



078 
576 
074 
572 
070 
567 
065 
562 
060 
557 



504 

504 

504 

503 

504 

503 

503 

503 

502 

503 

502 

503 

502 

502 

502 

501 

502 

501 

501 

502 

500 

501 

501 

501 

500 

500 

500 

500 

500 

500 

499 

500 

499 

499 

499 

499 

499 

498 

499 

498 

498 

498 

498 

498 

497 

498 

497 

498 

497 

497 



9.935 368 
9.935 497 
9.935 627 
9.935 756 

9.935 885 

9.936 013 
9.936 142 
9.936 270 
9.936 397 
9.936 525 
9.936 652 
9.936 779 

9.936 906 

9.937 032 
9.937 159 
9.937 285 
9.937 410 
9.937 536 
9.937 661 
9.937 786 

9.937 911 

9.938 035 
9.938 160 
9.938 284 
9.938 407 
9.938 531 
9.938 654 
9.938 777 

9.938 900 

9.939 022 

9.939 144 
9.939 266 
9.939 388 
9.939 510 
9.939 631 
9.939 752 
9.939 873 

9.939 993 

9.940 113 
9.940 233 
9.940 353 
9.940 473 
9.940 592 
9.940 711 
9.940 830 

9.940 949 
9.941067 

9.941 185 
9.941 303 
9.941421 



129 

130 

129 

129 

128 

129 

128 

127 

128 

127 

127 

127 

126 

127 

126 

125 

126 

125 

125 

125 

124 

125 

124 

123 

124 

123 

123 

123 

122 

122 

122 

122 

122 

121 

121 

121 

120 

120 

120 

120 

120 

119 

119 

119 

119 

118 

118 

118 

118 

117 



312 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



!<>K ciwh J 



log sinh z 



log tanh x 



1.350 
1.351 
1.352 
1.353 
1.354 
1.355 
1.356 
1.357 
1.358 
1.359 
1.360 
1.361 
1.362 
1.363 
1.364 
1.365 
1.366 
1.367 
1.368 
1.369 
1.370 
1.371 
1.372 
1.373 
1.374 
1.375 
1.376 
1.377 
1.378 
1.379 
1.380 
1.381 
1.382 
1.383 
1.384 
1.385 
1.386 
1.387 
1.388 
1.389 
1.390 
1.391 
1.392 
1.393 
1.394 
1.395 
1.396 
1.397 
1.398 
1.399 



0.313 516 
0.313 895 
0.314 275 
0.314 655 
0.315 035 
0.315 415 
0.315 795 
0.316 175 
0.316 556 
0.316 936 
0.317 317 
0.317 697 
0.318 078 
0.318 459 
0.318 840 
0.319 221 
0.319 602 
0.319 983 
0.320 365 
0.320 746 
0.321 128 
0.321 509 
0.321 891 
0.322 273 
0.322 655 
0.323 037 
0.323 419 
0.323 802 
0.324 184 
0.324 566 
0.324 949 
0.325 331 
0.325 714 
0.326 097 
0.326 480 
0.326 863 
0.327 246 
0.327 629 
0.328 013 
0.328 396 



0.32S 
0.329 
0.329 
0.329 
0.330 
0.330 
0.331 
0.331 
0.331 
0.332 



780 
163 
547 
931 
315 
699 
083 
467 
851 
235 



379 
380 
380 
380 
380 
380 
380 
381 
380 
381 
380 
381 
381 
381 
381 
381 
381 
382 
381 
382 
381 
382 
382 
382 
382 
382 
383 
382 
382 
383 
382 
383 
383 
383 
383 
383 
383 
384 
383 
384 
383 
384 
384 
384 
384 
384 
384 
384 
384 
385 



0.255 054 
0.255 550 
0.256 047 
0.256 544 
0.257 040 
0.257 536 
0.258 032 
0.258 528 
0.259 024 
0.259 520 
0.260 016 
0.260 511 
0.261006 
0.261 502 
0.261 997 
0.262 492 
0.262 987 
0.263 481 
0.263 976 
0.264 470 
0.264 965 
0.265 459 
0.265 953 
0.266 447 
0.266 941 
0.267 434 
0.267 928 
0.268 421 
0.268 915 
0.269 408 
0.269 901 
0.270 394 
0.270 886 
0.271 379 
0.271 872 
0.272 364 
0.272 856 
0.273 349 
0.273 841 
0.274 333 
0.274 824 
0.275 316 
0.275 808 
0.276 299 
0.276 790 
0.277 282 
0.277 773 
0.278 264 
0.278 754 
0.279 245 



496 
497 
497 
496 
496 
496 
496 
496 
496 
496 
495 
495 
496 
495 
495 
495 
494 
495 
494 
495 
494 
494 
494 
494 
493 
494 
493 
494 
493 
493 
493 
492 
493 
493 
492 
492 
493 
492 
492 
491 
492 
492 
491 
491 
492 
491 
491 
490 
491 
491 



9.941 538 
9.941 655 
9.941 772 

9.941 889 

9.942 005 
9.942 121 
9.942 237 
0.942 353 
9.942 469 
9.942 584 
9.942 699 
9.942 814 

9.942 928 

9.943 043 
9.943 157 
9.943 271 
9.943 384 
9.943 498 
9.943 611 
9.943 724 
9.943 837 

9.943 949 

9.944 062 
9.944 174 
9.944 286 
9.944 397 
9.944 509 
9.944 620 
9.944 731 
9.944 841 

9.944 952 

9.945 062 
9.945 172 
9.945 282 
9.945 392 
9.945 501 
9.945 610 
9.945 719 
9.945 828 

9.945 936 

9.946 045 
9.946 153 
9.946 261 
9.946 368 
9.946 476 
9.946 583 
9.946 690 
9.946 797 

9.946 903 

9.947 010 



17 
17 
17 
If, 
10 
16 
16 
16 
15 
15 
15 
11 
15 
14 
14 
13 
14 
13 
13 
13 
12 
13 
12 
12 
11 
12 
11 
11 
10 
11 
10 
10 
10 
10 
09 
09 
08 

09 

06 

09 
OS 
OS 
07 
OS 
07 
07 
07 
06 
07 
06 



APPENDIX 



313 



TABLED OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh x 



log tanh x 



1.400 
1.401 
1.402 
1.403 
1.404 
1.405 
1.406 
1.407 
1.408 
1.409 
1.410 
1.411 
1.412 
1.413 
1.414 
1.415 
1.416 
1.417 
1.418 
1.419 
1.420 
1.421 
1.422 
1.423 
1.424 
1.425 
1.426 
1.427 
1.428 
1.429 
1.430 
1.431 
1.432 



433 
434 
435 
436 
437 
438 
439 



1.440 
1.441 
1.442 
1.443 
1.444 
1.445 
1.446 
1.447 
1.448 
1.449 



0.332 620 
0.333 004 
0.333 389 
0.333 774 
0.334 159 
0.334 544 
0.334 929 
0.335 314 
0.335 699 
0.336 084 
C. 336 470 
C. 336 855 
C. 337 241 
C. 337 626 
0.338 012 
0.338 398 
0.338 784 
0.339 170 
0.339 556 
0.339 942 
0.340 329 
0.340 715 
0.341 101 
0.341 488 
0.341 875 
0.342 261 
0.342 648 
0.343 035 
0.343 422 
0.343 809 
0.344 197 
0.344 584 
0.344 971 
0.345 359 
0.345 746 
0.346 134 
0.346 522 
0.346 909 
0.347 297 
0.347 685 
0.348 073 
0.348 462 
0.348 850 
0.349 238 
0.349 627 
0.350 015 
0.350 404 
0.350 792 
0.351 181 
0.351 570 



384 
385 
385 
385 
385 
385 
385 
385 
385 
386 
385 
386 
385 
386 
386 
386 
386 
386 
386 
387 
386 
386 
387 
387 
386 
387 
387 
387 
387 
388 
387 
387 
388 
387 
388 
388 
387 
388 
388 
388 
389 
388 
388 
389 
388 
389 
388 
389 
389 
389 



0.279 736 
0.280 226 
0.280 717 
0.281 207 
0.281 697 
0.282 187 
0.282 677 
0.283 167 
0.283 656 
0.284 146 
0.284 635 
0.285 124 
0.285 614 
0.286 103 
0.286 592 
0.287 080 
0.287 569 
0.288 058 
0.288 546 
0.289 035 
0.289 523 
0.290 011 
0.290 499 
0.290 987 
0.291 475 
0.291 962 
0.292 450 
0.292 937 
0.293 425 
0.293 912 
0.294 399 
0.294 886 
0.295 373 
0.295 860 
0.296 346 
0.296 833 
0.297 319 
0.297 806 
0.298 292 
0.298 778 
0.299 264 
0.299 750 
0.300 236 
0.300 721 
0.301 207 
0.301 692 
0.302 178 
0.302 663 
0.303 148 
0.303 633 



490 
491 
490 
490 
490 
490 
490 
489 
490 
489 
489 
490 
489 
489 
488 
489 
489 
488 
489 
488 
488 
488 
488 
488 
487 
488 
487 
488 
487 
487 
487 
487 
487 
486 
487 
486 
487 
486 
486 
486 
486 
486 
485 
486 
485 
486 
485 
485 
485 
485 



9.947 116 
9.947 222 
9.947 327 
9.947 433 
9.947 538 
9.947 643 
9.947 748 
9.947 853 

9.947 957 

9.948 061 
9.948 165 
9.948 269 
9.948 373 
9.948 476 
9.948 579 
9.948 682 
9.948 785 
9.948 888 

9.948 990 

9.949 092 
9.949 194 
9.949 296 
9.949 398 
9.949 499 
9.949 600 
9.949 701 
9.949 802 

9.949 902 

9.950 002 
9.950 103 
9.950 202 
9.950 302 
9.950 402 
9.950 501 
9.950 600 
9.950 699 
9.950 798 
9.950 896 

9.950 995 
9.951093 

9.951 191 
9.951288 
9.951 386 
9.951483 
9.951580 
9.951 677 
9.951774 
9.951871 

9.951 967 

9.952 063 



106 

105 

106 

105 

105 

105 

105 

104 

104 

104 

104 

104 

103 

103 

103 

103 

103 

102 

102 

102 

102 

102 

101 

101 

101 

101 

100 

100 

101 

99 

100 

100 

99 

99 

99 

99 

98 

99 

98 

98 

97 

98 

97 

97 

97 

97 

97 

96 

96 

96 



314 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log ooflh z 



log sinh x 



log tanh z 



1.450 
1.451 



.452 
.453 
.454 
.455 
.456 
.457 
.458 
.459 



.460 
.461 
.462 
.463 
.464 
.465 
1.466 
1.467 
1.468 
1.469 
1.470 
1.471 
1.472 
1.473 
1.474 
1.475 
1.476 
1.477 
1.478 
1.479 
1.480 
1.481 
1.482 
1.483 
1.484 
1.485 
1.486 
1.487 
1.488 
1.489 
1.490 
1.491 
1.492 
1.493 
1.494 
1.495 
1.496 
1.497 
1.498 
1.499 



0.351959 
0.352 348 
0.352 737 
0.353 126 
0.353 516 
0.353 905 
0.354 295 
0.354 684 
0.355 074 
0.355 463 
0.355 853 
0.356 243 
0.356 633 
0.357 023 
0.357 413 
0.357 804 
0.358 194 
0.358 584 
0.358 975 
0.359 365 
0.359 756 
0.360 147 
0.360 537 
0.360 928 
0.361 319 
0.361 710 
0.362 101 
0.362 493 
0.362 884 
0.363 275 
0.363 667 
0.364 058 
0.364 450 
0.364 842 
0.365 233 
0.365 625 
0.366 017 
0.366 409 
0.366 801 
0.367 194 
0.367 586 
0.367 978 
0.368 371 
0.368 763 
0.369 156 
0.369 548 
0.369 941 
0.370 334 
0.370 727 
0.371 120 



389 
389 
389 
390 
389 
390 
389 
390 
389 
390 
390 
390 
390 
390 
391 
390 
390 
391 
390 
391 
391 
390 
391 
391 
391 
391 
392 
391 
391 
392 

391 
392 
392 
391 
392 
392 
392 
392 
393 
392 
392 
393 
392 
393 
392 
393 
393 
393 
393 
393 



0.304 118 
0.304 603 
0.305 088 
0.305 572 
0.306 057 
0.306 541 
0.307 025 
0.307 510 
0.307 994 
0.308 478 
0.308 962 
0.309 445 
0.309 929 
0.310 412 
0.310 896 
0.311 379 
0.311 863 
0.312 346 
0.312 829 
0.313 312 
0.313 794 
0.314 277 
0.314 760 
0.315 242 
0.315 725 
0.316 207 
0.316 689 
0.317 171 
0.317 653 
0.318 135 
0.318 617 
0.319 099 
0.319 580 
0.320 062 
0.320 543 
0.321 025 
0.321 506 
0.321 987 
0.322 468 
0.322 949 
0.323 430 
0.323 911 
0.324 391 
0.324 872 
0.325 352 
0.325 832 
0.326 313 
0.326 793 
0.327 273 
0.327 753 



485 
485 
484 
485 
484 
484 
485 
484 
484 
484 
483 
484 
483 
484 
483 
484 
483 
483 
483 
482 
483 
483 
482 
483 
482 
482 
482 
482 
482 
482 
482 
481 
482 
481 
482 
481 
481 
481 
481 
481 
481 
480 
481 
480 
480 
481 
480 
480 
480 
480 



9.952 159 
9.952 255 
9.952 350 
9.952 446 
9.952 541 
9.952 636 
9.952 731 
9.952 826 

9.952 920 

9.953 014 
9.953 108 
9.953 202 
9.953 296 
9.953 389 
9.953 483 
9.953 576 
9.953 669 
9.953 761 
9.953 854 

9.953 946 

9.954 039 
9.954 131 
9.954 222 
9.954 314 
9.954 405 
9.954 497 
9.954 588 
9.954 679 
9.954 769 
9.954 860 

9.954 950 

9.955 040 
9.955 130 
9.955 220 
9.955 310 
9.955 399 
9.955 489 
9.955 578 
9.955 667 
9.955 755 
9.955 844 

9.955 932 

9.956 020 
9.956 109 
9.956 196 
9.956 284 
9.956 372 
9.956 459 
9.956 546 
9.956 633 



96 
95 
96 
95 
95 
95 
95 
94 
94 
94 
94 
94 
93 
94 
93 
93 
92 
93 
92 
93 
92 
91 
92 
91 
92 
91 
91 
90 
91 
90 
90 
90 
90 
90 
89 
90 
89 
89 
88 
89 
88 
88 
89 
87 
88 
88 
87 
87 
87 
87 



APPENDIX 



315 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



1.500 

1.501 

1.502 

1.503 

1.504 

1.505 

1.506 

1.507 

1.508 

1.509 

1.510 

1.511 

1.512 

1.513 

1.514 

1.515 

1.516 

1.517 

1.518 

1.519 

1.520 

1.521 

1.522 

1.523 

1.524 

1.525 

1.526 

1.527 

1.528 

1.529 

1.530 

1.531 

1.532 

1.533 

1.534 

1.535 

1.536 

1.537 

1.538 

1.539 

1.540 

1.541 

1.542 

1.543 

1.544 

1.545 

1.546 

1.547 

1.548 

1.549 



log cosh x 



0.371513 
0.371906 
0.372 299 
0.372 693 
0.373 086 
0.373 479 
0.373 873 
0.374 267 
0.374 660 
0.375 054 
0.375 448 
0.375 842 
0.376 236 
0.376 630 
0.377 024 
0.377 418 
0.377 813 
0.378 207 
0.378 601 
0.378 996 
0.379 390 
0.379 785 
0.380 180 
0.380 575 
0.380 970 
0.381365 
0.381 760 
0.382 155 
0.382 550 
0.382 945 
0.383 341 
0.383 736 
0.384 132 
0.384 527 
0.384 923 
0.385 319 
0.385 714 
0.386 110 
0.386 506 
0.386 902 
0.387 298 
0.387 694 
0.388 091 
0.388 487 
0.388 883 
0.389 280 
0.389 676 
0.390 073 
0.390 470 
0.390 866 



393 

393 

394 

393 

393 

394 

394 

393 

394 

394 

394 

394 

394 

394 

394 

395 

394 

394 

395 

394 

395 

395 

395 

395 

395 

395 

395 

395 

395 

396 

395 

396 

395 

396 

396 

395 

396 

396 

396 

396 

396 

397 

396 

396 

397 

396 

397 

397 

396 

397 



log sinh x 



0.328 233 
0.328 712 
0.329 192 
0.329 672 
0.330 151 
0.330 630 
0.331 110 
0.331 589 
0.332 068 
0.332 547 
0.333 026 
0.333 505 
0.333 983 
0.334 462 
0.334 941 
0.335 419 
0.335 897 
0.336 376 
0.336 854 
0.337 332 
0.337 810 
0.338 288 
0.338 766 
0.339 243 
0.339 721 
0.340 198 
0.340 676 
0.341 153 
0.341 630 
0.342 108 
0.342 585 
0.343 062 
0.343 539 
0.344 015 
0.344 492 
0.344 969 
0.345 445 
0.345 922 
0.346 398 
0.346 874 
0.347 350 
0.347 827 
0.348 303 
0.348 778 
0.349 254 
0.349 730 
0.350 206 
0.350 681 
0.351 157 
0.351 632 



479 

480 

480 

479 

479 

480 

479 

479 

479 

479 

479 

478 

479 

479 

478 

478 

479 

478 

478 

478 

478 

478 

477 

478 

477 

478 

477 

477 

478 

477 

477 

477 

476 

477 

477 

476 

477 

476 

476 

476 

477 

476 

475 

476 

476 

476 

475 

476 

475 

475 



log tanh z 



9.956 720 
9.956 806 
9.956 893 

9.956 979 

9.957 065 
9.957 151 
9.957 237 
9.957 322 
9.957 408 
9.957 493 
9.957 578 
9.957 663 
9.957 748 
9.957 832 

9.957 917 

9.958 001 
9.958 085 
9.958 169 
9.958 252 
9.958 336 
9.958 419 
9.958 503 
9.958 586 
9.958 669 
9.958 751 
9.958 834 
9.958 916 

9.958 998 

9.959 080 
9.959 162 
9.959 244 
9.959 326 
9.959 407 
9.959 488 
9.959 569 
9.959 650 
9.959 731 
9.959 811 
9.959 892 

9.959 972 

9.960 052 
9.960 132 
9.960 212 
9.960 291 
9.960 371 
9.960 450 
9.960 529 
9.960 608 
9.960 687 
9.960 766 



87 



86 
86 
85 
86 
85 
85 
85 
85 
84 
85 
84 
84 
84 
83 
84 
83 
84 
83 
83 
82 
83 
82 
82 
82 
82 
82 
82 
81 
81 
81 
81 
81 
80 
81 
80 
80 
80 
80 
79 
80 
79 
79 
79 
79 
79 
78 



316 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh z 



log sinh x 



1(>k tanli z 



1.550 
1.551 
1.552 
1.553 
1.554 
1.555 
1.556 
1.557 
1.558 
1.559 
1.560 
1.561 
1.562 
1.563 
1.564 
1.565 
1.566 
1.567 
1.568 
1.569 



570 
571 
572 
573 
574 
1.575 
1.576 
1.577 
1.578 
1.579 
1.580 
1.581 
1.582 
1.583 
1.584 
1.585 
1.586 
1.587 
1.588 
1.589 
1.590 
1.591 
1.592 
1.593 
1.594 
1.595 
1.596 
1.597 
1.598 
1.599 



0.391 263 
0.391 660 
0.392 057 
0.392 454 
0.392 851 
0.393 248 
0.393 646 
0.394 043 
0.394 440 
0.394 838 
0.395 235 
0.395 633 
0.396 031 
0.396 428 
0.396 826 
0.397 224 
0.397 622 
0.398 020 
0.398 418 
0.398 816 
0.399 214 
0.399 613 
0.400 011 
0.400 409 
0.400 808 
0.401 207 
0.401 605 
0.402 004 
0.402 403 
0.402 802 

0.403 200 
0.403 599 
0.403 998 
0.404 398 
0.404 797 
0.405 196 
0.405 595 
0.405 995 
0.406 394 
0.406 794 
0.407 193 
0.407 593 
0.407 993 
0.408 392 
0.408 792 
0.409 192 
0.409 592 
0.409 992 
0.410 392 
0.410 792 



397 
397 
397 
397 
397 
398 
397 
397 
398 
397 
398 
398 
397 
398 
398 
398 
398 
398 
398 
398 
399 
398 
398 
399 
399 
398 
399 
399 
399 
398 
399 
399 
400 
399 
399 
399 
400 
399 
400 
399 
400 
400 
399 
400 
400 
400 
400 
400 
400 
401 



0.352 107 
0.352 583 
0.353 058 
0.353 533 
0.354 008 
0.354 483 
0.354 958 
0.355 432 
0.355 907 
0.356 381 
0.356 856 
0.357 330 
0.357 805 
0.358 279 
0.358 753 
0.359 227 
0.359 701 
0.360 175 
0.360 649 
0.361 122 
0.361 596 
0.362 069 
0.362 543 
0.363 016 
0.363 490 
0.363 963 
0.364 436 
0.364 909 
0.365 382 
0.365 855 
0.366 328 
0.366 800 
0.367 273 
0.367 746 
0.368 218 
0.368 691 
0.369 163 
0.369 635 
0.370 107 
0.370 580 
0.371 052 
0.371 523 
0.371 995 
0.372 467 
0.372 939 
0.373 410 
0.373 882 
0.374 353 
0.374 825 
0.375 296 



476 
475 
475 
475 
475 
475 
474 
475 
474 
475 
474 
475 
474 
474 
474 
474 
474 
474 
473 
474 
473 
474 
473 
474 
473 
473 
473 
473 
473 
473 
472 
473 
473 
472 
473 
472 
472 
472 
473 
472 
471 
472 
472 
472 
471 
472 
471 
472 
471 
471 



9.960 844 

9.960 923 
9.961001 
9.961079 

9.961 157 
9.961234 
9.961312 
9.961 389 
9.961 467 
9.961544 
9.961621 
9.961697 
9.961774 
9.961850 

9.961 927 

9.962 003 
9.962 079 
9.962 155 
9.962 231 
9.962 306 
9.962 382 
9.962 457 
9.962 532 
9.962 607 
9.962 682 
9.962 756 
9.962 831 
9.962 905 

9.962 979 

9.963 053 
9.963 127 
9.963 201 
9.963 275 
9.963 348 
9.963 421 
9.963 495 
9.963 568 
9.963 641 
9.963 713 
9.963 786 
9.963 858 

9.963 931 

9.964 003 
9.964 075 
9.964 147 
9.964 218 
9.964 290 
9.964 361 
9.964 433 
9.964 504 



APPENDIX 



317 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh x 



log sinh z 



log tanh x 



1.600 
1.601 
1.602 
1.603 
1.604 
1.605 
1.606 
1.607 
1.608 
1.609 
1.610 
1.611 
1.612 
1.613 
1.614 
1.615 
1.616 
1.617 
1.618 
1.619 
1.620 



.621 
.622 
.623 
.624 
.625 
1.626 
1.627 
1.628 
1.629 
1.630 
1.631 
1.632 
1.633 
1.634 
1.635 
1.636 
1.637 
1.638 
1.639 
1.640 
1.641 
1.642 
1.643 
1.644 
1.645 
1.646 
1.647 
1.648 
1.649 



0.411 193 
0.411 593 
0.411 993 
0.412 394 
0.412 794 
0.413 195 
0.413 596 
0.413 996 
0.414 397 
0.414 798 
0.415 199 
0.415 600 
0.416 001 
0.416 402 
0.416 803 
0.417 204 
0.417 605 
0.418 007 
0.418 408 
0.418 810 
0.419 211 
0.419 613 
0.420 014 
0.420 416 
0.420 818 
0.421 220 
0.421 622 
0.422 024 
0.422 426 
0.422 828 
0.423 230 
0.423 632 
0.424 034 
0.424 437 
0.424 839 
0.425 242 
0.425 644 
0.426 047 
0.426 449 
0.426 852 
0.427 255 
0.427 658 
0.428 060 
0.428 463 
0.428 866 
0.429 269 
0.429 673 
0.430 076 
0.430 479 
0.430 882 



400 
400 
401 
400 
401 
401 
400 
401 
401 
401 
401 
401 
401 
401 
401 
401 
402 
401 
402 
401 
402 
401 
402 
402 
402 
402 
402 
402 
402 
402 
402 
402 
403 
402 
403 
402 
403 
402 
403 
403 
403 
402 
403 
403 
403 
404 
403 
403 
403 
404 



0.375 767 
0.376 239 
0.376 710 
0.377 181 
0.377 652 
0.378 123 
0.378 593 
0.379 064 
0.379 535 
0.380 005 
0.380 476 
0.380 946 
0.381 416 
0.381 887 
0.382 357 
0.382 827 
0.383 297 
0.383 767 
0.384 237 
0.384 707 
0.385 176 
0.385 646 
0.386 116 
0.386 585 
0.387 055 
0.387 524 
0.387 993 
0.388 462 
0.388 932 
0.389 401 
0.389 870 
0.390 339 
0.390 807 
0.391 276 
0.391 745 
0.392 214 
0.392 682 
0.393 151 
0.393 619 
0.394 088 
0.394 556 
0.395 024 
0.395 492 
0.395 960 
0.396 428 
0.396 896 
0.397 364 
0.397 832 
0.398 300 
0.398 767 



472 
471 
471 
471 
471 
470 
471 
471 
470 
471 
470 
470 
471 
470 
470 
470 
470 
470 
470 
469 
470 
470 
469 
470 
469 
469 
469 
470 
469 
469 
469 
468 
469 
469 
469 
468 
469 
468 
469 
468 
468 
468 
468 
468 
468 
468 
468 
468 
467 
468 



9.964 575 
9.964 646 
9.964 716 
9.964 787 
9.964 857 
9.964 928 

9.964 998 

9.965 068 
9.965 138 
9.965 207 
9.965 277 
9.965 346 
9.965 416 
9.965 485 
9.965 554 
9.965 623 
9.965 692 
9.965 760 
9.965 829 
9.965 897 

9.965 965 

9.966 033 
9.966 101 
9.966 169 
9.966 237 
9.966 304 
9.966 372 
9.966 439 
9.966 506 
9.966 573 
9.966 640 
9.966 707 
9.966 773 
9.966 840 
9.966 906 

9.966 972 

9.967 038 
9.967 104 
9.967 170 
9.967 236 
9.967 301 
9.967 366 
9.967 432 
9.967 497 
9.967 562 
9.967 627 
9.967 691 
9.967 756 
9.967 821 
9.967 885 



71 
70 
71 
70 
71 
70 
70 
70 
69 
70 
69 
70 
69 
69 
69 
69 
68 
69 
68 
68 
6? 
68 
68 
68 
67 
68 
67 
67 
67 
67 
67 
66 
67 
66 
66 
66 
66 
66 
66 
65 
65 
66 
65 
65 
65 
64 
65 
65 
64 
64 



318 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



1.650 
1.651 
1.652 
1.653 
1.654 
1.655 
1.656 
1.657 
1.658 
1.659 
1.660 
1.661 
1.662 
1.663 
1.664 
1.665 
1.666 
1.667 
1.668 
1.669 
1.670 
1.671 
1.672 
1.673 
1.674 
1.675 
1.676 
1.677 
1.678 
1.679 
1.680 
1.681 
1.682 
1.683 
1.684 
1.685 
1.686 
1.687 
1.688 
1.689 
1.690 
1.691 
1.692 
1.693 
1.694 
1.695 
1.696 
1.697 
1.698 
1.699 



1«>C n.-h i 



0.431286 
0.431 689 
0.432 093 
0.432 496 
0.432 900 
0.433 303 
0.433 707 
0.434 111 
0.434 515 
0.434 919 
0.435 323 
0.435 727 
0.436 131 
0.436 535 
0.436 939 
0.437 343 
0.437 748 
0.438 152 
0.438 556 
0.438 961 
0.439 365 
0.439 770 
0.440 175 
0.440 579 
0.440 984 
0.441 389 
0.441 794 
0.442 199 
0.442 604 
0.443 009 
0.443 414 
0.443 819 
0.444 224 
0.444 630 
0.445 035 
0.445 440 
0.445 846 
0.446 251 
0.446 657 
0.447 062 
0.447 468 
0.447 874 
0.448 280 
0.448 685 
0.449 091 
0.449 497 
0.449 903 
0.450 309 
0.450 715 
0.451 122 



403 
404 
403 
404 
403 
404 
404 
404 
404 
404 
404 
404 
404 
404 
404 
405 
404 
404 
405 
404 
405 
405 
404 
405 
405 
405 
405 
405 
405 
405 
405 
405 
406 
405 
405 
406 
405 
406 
405 
406 
406 
406 
405 
406 
406 
406 
406 
406 
407 
406 



log sinh z 



0.399 235 
0.399 702 
0.400 170 
0.400 637 
0.401 105 
0.401 572 
0.402 039 
0.402 506 
0.402 973 
0.403 440 
0.403 907 
0.404 374 
0.404 841 
0.405 307 
0.405 774 
0.406 241 
0.406 707 
0.407 173 
0.407 640 
0.408 106 
0.408 572 
0.409 039 
0.409 505 
0.409 971 
0.410 437 
0.410 903 
0.411 368 
0.411 834 
0.412 300 
0.412 766 
0.413 231 
0.413 697 
0.414 162 
0.414 628 
0.415 093 
0.415 558 
0.416 023 
0.416 488 
0.416 954 
0.417 419 
0.417 883 
0.418 348 
0.418 813 
0.419 278 
0.419 743 
0.420 207 
0.420 672 
0.421 136 
0.421 601 
0.422 065 



467 
468 
467 
468 
467 
467 
467 
467 
467 
467 
467 
467 
466 
467 
467 
466 
466 
467 
466 
466 
467 
466 
466 
466 
466 
465 
466 
466 
466 
465 
466 
465 
466 
465 
465 
465 
465 
466 
465 
464 
465 
465 
465 
465 
464 
465 
464 
465 
464 
464 



log tanli i 



9.967 949 

9.968 013 
9.968 077 
9.968 141 
9.968 205 
9.968 268 
9.968 332 
9.968 395 
9.968 458 
9.968 521 
9.968 584 
9.968 647 
9.968 710 
9.968 772 
9.968 835 
9.968 897 

9.968 959 

9.969 021 
9.969 083 
9.969 145 
9.969 207 
9.969 269 
9.969 330 
9.969 391 
9.969 453 
9.969 514 
9.969 575 
9.969 635 
9.969 696 
9.969 757 
9.969 817 
9.969 878 
9.969 938 

9.969 998 

9.970 058 
9.970 118 
9.970 178 
9.970 237 
9.970 297 
9.970 356 
9.970 415 
9.970 474 
9.970 534 
9.970 592 
9.970 651 
9.970 710 
9.970 768 
9.970 827 
9.970 885 
9.970 943 



64 
64 
64 
64 
63 
64 
63 
63 
63 
63 
63 
63 
62 
63 
62 
62 
62 
62 
62 
62 
62 
61 
61 
62 
61 
61 
60 
61 
61 
60 
61 
60 
60 
60 
60 
60 
59 
'60 
59 
59 
59 
60 
58 
59 
59 
58 
59 
58 
58 
59 



APPENDIX 



319 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh z 



log sinh / 



log tanh i 



1.700 
1.701 
1.702 
1.703 
1.704 
1.705 
1.706 
1.707 
1.708 
1.709 
1.710 
1.711 
1.712 
1.713 
1.714 
1.715 
1.716 
1.717 
1.718 
1.719 
1.720 
1.721 
1.722 
1.723 
1.724 
1.725 
1.726 
1.727 
1.728 
1.729 
1.730 
1.731 
1.732 
1.733 
1.734 
1.735 
1.736 
1.737 
1.738 
1.739 
1.740 
1.741 
1.742 
1.743 
1.744 
1.745 
1.746 
1.747 
1.748 
1.749 



0.451 528 
0.451934 
0.452 340 
0.452 747 
0.453 153 
0.453 560 
0.453 966 
0.454 373 
0.454 780 
0.455 186 
0.455 593 
0.456 000 
0.456 407 
0.456 814 
0.457 221 
0.457 628 
0.458 035 
0.458 442 
0.458 849 
0.459 256 
0.459 663 
0.460 071 
0.460 478 
0.460 886 
0.461 293 
0.461 701 
0.462 108 
0.462 516 
0.462 924 
0.463 331 
0.463 739 
0.464 147 
0.464 555 
0.464 963 
0.465 371 
0.465 779 
0.466 187 
0.466 595 
0.467 003 
0.467 412 

0.467 820 
0.468 228 
0.468 637 
0.469 045 
0.469 454 
0.469 862 
0.470 271 
0.470 680 
0.471 088 
0.471 497 



406 
406 
407 
406 
407 
406 
407 
407 
406 
407 
407 
407 
407 
407 
407 
407 
407 
407 
407 
407 
408 
407 
408 
407 
408 
407 
408 
408 
407 
408 
408 
408 
408 
408 
408 
408 
408 
408 
409 
408 

408 
409 
408 
409 
408 
409 
409 
408 
409 
409 



0.422 529 
0.422 994 
0.423 458 
0.423 922 
0.424 386 
0.424 850 
0.425 314 
0.425 778 
0.426 242 
0.426 705 
0.427 169 
0.427 633 
0.428 096 
0.428 560 
0.429 023 
0.429 487 
0.429 950 
0.430 413 
0.430 877 
0.431 340 
0.431 803 
0.432 266 
0.432 729 
0.433 192 
0.433 655 
0.434 117 
0.434 580 
0.435 043 
0.435 506 
0.435 968 
0.436 431 
0.436 893 
0.437 355 
0.437 818 
0.438 280 
0.438 742 
0.439 204 
0.439 667 
0.440 129 
0.440 591 

0.441052 
0.441514 
0.441976 
0.442 438 
0.442 900 
0.443 361 
0.443 823 
0.444 284 
0.444 746 
0.445 207 



465 
464 
464 
464 
464 
464 
464 
464 
463 
464 
464 
463 
464 
463 
464 
463 
463 
464 
463 
463 
463 
463 
463 
463 
462 
463 
463 
463 
462 
463 
462 
462 
463 
462 
462 
462 
463 
462 
462 
461 

462 
462 
462 
462 
461 
462 
461 
462 
461 
462 



9.971 002 
9.971060 
9.971 117 
9.971 175 
9.971233 
9.971 290 
9.971 348 
9.971405 
9.971 462 
9.971519 
9.971 576 
9.971633 
9.971 690 
9.971 746 
9.971803 
9.971 859 
9.971915 

9.971 972 

9.972 028 
9.972 084 
9.972 139 
9.972 195 
9.972 251 
9.972 306 
9.972 362 
9.972 417 
9.972 472 
9.972 527 
9.972 582 
9.972 637 
9.972 691 
9.972 746 
9.972 801 
9.972 855 
9.972 909 

9.972 963 

9.973 017 
9.973 071 
9.973 125 
9.973 179 
9.973 233 
9.973 286 
9.973 339 
9.973 393 
9.973 446 
9.973 499 
9.973 552 
9.973 605 
9.973 658 
9.973 710 



58 
57 
58 
58 
57 
58 
57 
57 
57 
57 
57 
57 
56 
57 
56 
56 
57 
56 
56 
55 
56 
56 
55 
56 
55 
55 
55 
55 
55 
54 
55 
55 
54 
54 
54 
54 
54 
54 
54 
54 
53 
'53 
54 
53 
53 
53 
53 
53 
52 
53 



320 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



1<>K cosh i 



log sinh z 



log tanh z 



1.750 


0.471 906 


1.751 


0.472 315 


1.752 


0.472 724 


1.753 


0.473 133 


1.754 


0.473 542 


1.755 


0.473 951 


1.756 


0.474 360 


1.757 


0.474 769 


1.758 


0.475 178 


1.759 


0.475 587 


1.760 


0.475 997 


1.761 


0.476 406 


1.762 


0.476 815 


1.763 


0.477 225 


1.764 


0.477 634 


1.765 


0.478 044 


1.766 


0.478 453 


1.767 


0.478 863 


1.768 


0.479 273 


1.769 


0.479 682 


1.770 


0.480 092 


1.771 


0.480 502 


1.772 


0.480 912 


1.773 


0.481322 


1.774 


0.481 732 


1.775 


0.482 142 


1.776 


0.482 552 


1.777 


0.482 962 


1.778 


0.483 372 


1.779 


0.483 782 


1.780 


0.484 193 


1.781 


0.484 603 


1.782 


0.485 013 


1.783 


0.485 424 


1.784 


0.485 834 


1.785 


0.486 245 


1.786 


0.486 655 


1.787 


0.487 066 


1.788 


0.487 476 


1.789 


0.487 887 


1.790 


0.488 298 


1.791 


0.488 708 


1.792 


0.489 119 


1.793 


0.489 530 


1.794 


0.489 941 


1.795 


0.490 352 


1.796 


0.490 763 


1.797 


0.491 174 


1.798 


0.491 585 


1.799 


0.491 996 



409 

409 

409 

409 

409 

409 

409 

409 

409 

410 

409 

409 

410 

409 

410 

409 

410 

410 

409 

410 

410 

410 

410 

410 

410 

410 

410 

410 

410 

411 

410 

410 

411 

410 

411 

410' 

411 

410 

411 

411 

410 

411 

411 

411 

411 

411 

411 

411 

411 

411 



0.445 669 
0.446 130 
0.446 591 
0.447 052 
0.447 514 
0.447 975 
0.448 436 
0.448 897 
0.449 358 
0.449 819 
0.450 279 
0.450 740 
0.451 201 
0.451661 
0.452 122 
0.452 583 
0.453 043 
0.453 504 
0.453 964 
0.454 424 
0.454 885 
0.455 345 
0.455 805 
0.456 265 
0.456 725 
0.457 185 
0.457 645 
0.458 105 
0.458 565 
0.459 025 
0.459 484 
0.459 944 
0.460 404 
0.460 863 
0.461 323 
0.461 782 
0.462 242 
0.462 701 
0.463 160 
0.463 620 
0.464 079 
0.464 538 
0.464 997 
0.465 456 
0.465 915 
0.466 374 
0.466 833 
0.467 292 
0.467 751 
0.468 210 



461 
461 
461 
462 
461 
461 
461 
461 
461 
460 
461 
461 
460 
461 
461 
460 
461 
460 
460 
461 
460 
460 
.460 
460 
460 
460 
460 
460 
460 
459 
460 
460 
459 
460 
459 
460 
459 
459 
460 
459 
459 
459 
459 
459 
459 
459 
459 
459 
459 
458 



9.973 763 
9.973 815 
9.973 868 
9.973 920 

9.973 972 

9.974 024 
9.974 076 
9.974 128 
9.974 180 
9.974 231 
9.974 283 
9.974 334 
9.974 385 
9.974 437 
9.974 488 
9.974 539 
9.974 590 
9.974 640 
9.974 691 
9.974 742 
9.974 792 
9.974 843 
9.974 893 
9.974 943 

9.974 993 

9.975 043 
9.975 093 
9.975 143 
9.975 193 
9.975 242 
9.975 292 
9.975 341 
9.975 390 
9.975 440 
9.975 489 
9.975 538 
9.975 587 
9.975 635 
9.975 684 
9.975 733 
9.975 781 
9.975 830 
9.975 878 
9.975 926 

9.975 974 

9.976 022 
9.976 070 
9.976 118 
9.976 166 
9.976 213 



52 
53 
52 
52 
52 
52 
52 
52 
51 
52 

51 
51 
52 
51 
51 
51 
50 
51 
51 
50 
51 
50 
50 
50 
50 
50 
50 
50 
49 
50 
49 
49 
50 
49 
49 
49 
48 
49 
49 
48 
49 
48 
48 
48 
48 
48 
48 
48 
47 
48 



APPENDIX 



321 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh z 



log si nil X 



log tanh x 



1.800 
1.801 
1.802 
1.803 
1.804 
1.805 
1.806 
1.807 
1.808 
1.809 
1.810 
1.811 
1.812 
1.813 
1.814 
1.815 
1.816 
1.817 
1.818 
1.819 
1.820 
1.821 
1.822 
1.823 
1.824 
1.825 
1.826 
1.827 
1.828 
1.829 
1.830 
1.831 
1.832 
1.833 
1.834 
1.835 
1.836 
1.837 
1.838 
1.839 
1.840 
1.841 



.842 
.843 
.844 
.845 
.846 
.847 
1.848 
1.849 



0.492 407 
0.492 819 
0.493 230 
0.493 641 
0.494 053 
0.494 464 
0.494 875 
0.495 287 
0.495 698 
0.496 110 
0.496 522 
0.496 933 
0.497 345 
0.497 757 
0.498 168 
0.498 580 
0.498 992 
0.499 404 
0.499 816 
0.500 228 
0.500 640 
0.501 052 
0.501 464 
0.501 877 
0.502 289 
0.502 701 
0.503 113 
0.503 526 
0.503 938 
0.504 351 
0.504 763 
0.505 176 
0.505 588 
0.506 001 
0.506 413 
0.506 826 
0.507 239 
0.507 652 
0.508 064 
0.508 477 
0.508 890 
0.509 303 
0.509 716 
0.510 129 
0.510 542 
0.510 955 
0.511 368 
0.511 782 
0.512 195 
0.512 608 



412 
411 
411 
412 
411 
411 
412 
411 
412 
412 
411 
412 
412 
411 
412 
412 
412 
412 
412 
412 
412 
412 
413 
412 
412 
412 
413 
412 
413 
412 
413 
412 
413 
412 
413 
413 
413 
412 
413 
413 
413 
413 
413 
413 
413 
413 
414 
413 
413 
414 



0.468 668 
0.469 127 
0.469 586 
0.470 044 
0.470 503 
0.470 961 
0.471 420 
0.471 878 
0.472 336 
0.472 795 
0.473 253 
0.473 711 
0.474 169 
0.474 627 
0.475 085 
0.475 543 
0.476 001 
0.476 459 
0.476 917 
0.477 375 
0.477 832 
0.478 290 
0.478 748 
0.479 205 
0.479 663 
0.480 120 
0.480 578 
0.481 035 
0.481 493 
0.481 950 
0.482 407 
0.482 864 
0.483 321 
0.483 779 
0.484 236 
0.484 693 
0.485 150 
0.485 607 
0.486 064 
0.486 520 
0.486 977 
0.487 434 
0.487 891 
0.488 347 
0.488 804 
0.489 260 
0.489 717 
0.490 173 
0.490 630 
0.491 086 



459 
459 
458 
459 
458 
459 
458 
458 
459 
458 
458 
458 
458 
458 
458 
458 
458 
458 
458 
457 
458 
458 
457 
458 
457 
458 
457 
458 
457 
457 
457 
457 
458 
457 
457 
457 
457 
457 
456 
457 
457 
457 
456 
457 
456 
457 
456 
457 
456 
457 



9.976 261 
9.976 308 
9.976 356 
9.976 403 
9.976 450 
9.976 497 
9.976 544 
9.976 591 
9.976 638 
9.976 685 
9.976 731 
9.976 778 
9.976 824 
9.976 871 
9.976 917 

9.976 963 

9.977 009 
9.977 055 
9.977 101 
9.977 147 
9.977 192 
9.977 238 
9.977 283 
9.977 329 
9.977 374 
9.977 419 
9.977 464 
9.977 509 
9.977 554 
9.977 599 
9.977 644 
9.977 689 
9.977 733 
9.977 778 
9.977 822 
9.977 867 
9.977 911 
9.977 955 

9.977 999 

9.978 043 
9.978 087 
9.978 131 
9.978 174 
9.978 218 
9.978 262 
9.978 305 
9.978 348 
9.978 392 
9.978 435 
9.978 478 



47 
48 
47 
47 
47 
47 
47 
47 
47 
46 
47 
46 
47 
46 
46 
46 
46 
46 
46 
45 
46 
45 
46 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
44 
45 
44 
44 
44 
44 
44 
44 
43 
44 
44 
43 
43 
44 
43 
43 
43 



322 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Continued) 



log cosh z 



log hi ii h z 



log t null z 



1.850 
1.851 
1.S52 
1.853 
1.854 
1.855 
1.856 
1.857 
1.858 
1.859 
1.860 
1.861 
1.862 
1.863 
1.864 
1.865 
1.866 
1.867 
1.868 
1.869 
1.870 
1.871 
1.872 



873 
,874 
.875 
.876 
.877 
.878 
.879 



1.880 
1.881 
1.882 
1.883 
1.884 
1.885 
1.886 
1.887 
1.888 
1.889 
1.890 
1.891 
1.892 
1.893 
1.894 
1.895 
1.896 
1.897 
1.898 
1.899 



0.513 022 
0.513 435 
0.513 848 
0.514 262 
0.514 675 
0.515 089 
0.515 502 
0.515 916 
0.516 330 
0.516 743 
0.517 157 
0.517 571 
0.517 984 
0.518 398 
0.518 812 
0.519 226 
0.519 640 
0.520 054 
0.520 468 
0.520 882 
0.521 296 
0.521 711 
0.522 125 
0.522 539 
0.522 953 
0.523 368 
0.523 782 
0.524 196 
0.524 611 
0.525 025 
0.525 440 
0.525 854 
0.526 269 
0.526 683 
0.527 098 
0.527 513 
0.527 928 
0.528 342 
0.528 757 
0.529 172 
0.529 587 
0.530 002 
0.530 417 
0.530 832 
0.531 247 
0.531 662 
0.532 077 
0.532 492 
0.532 907 
0.533 323 



413 
413 
414 
413 
414 
413 
414 
414 
413 
414 
414 
413 
414 
414 
414 
414 
414 
414 
414 
414 
415 
414 
414 
414 
415 
414 
414 
415 
414 
415 
414 
415 
414 
415 
415 
415 
414 
415 
415 
415 
415 
415 
415 
415 
415 
415 
415 
415 
416 
415 



0.491 543 
0.491 999 
0.492 455 
0.492 911 
0.493 367 
0.493 824 
0.494 280 
0.494 736 
0.495 192 
0.495 648 
0.496103 
0.496 559 
0.497 015 
0.497 471 
0.497 927 
0.498 382 
0.498 838 
0.499 293 
0.499 749 
0.500 204 
0.500 660 
0.501 115 
0.501571 
0.502 026 
0.502 481 
0.502 937 
0.503 392 
0.503 847 
0.504 302 
0.504 757 
0.505 212 
0.505 667 
0.506 122 
0.506 577 
0.507 032 
0.507 487 
0.507 941 
0.508 396 
0.508 851 
0.509 305 
0.509 760 
0.510 215 
0.510 669 
0.511 124 
0.511 578 
0.512 032 
0.512 487 
0.512 941 
0.513 395 
0.513 850 



456 
456 
456 
456 
457 
456 
456 
456 
456 
455 
456 
456 
456 
456 
455 
456 
455 
456 
455 
456 
455 
456 
455 
455 
456 
455 
455 
455 
455 
455 
455 
455 
455 
455 
455 
454 
455 
455 
454 
455 
455 
454 
455 
454 
454 
455 
454 
454 
455 
454 



9.978 521 
9.978 564 
9.978 607 
9.978 650 
9.978 692 
9.978 735 
9.978 777 
9.978 820 
9.978 862 
9.978 904 
9.978 947 

9.978 989 

9.979 031 
9.979 073 
9.979 114 
9.979 156 
9.979 198 
9.979 239 
9.979 281 
9.979 322 
9.979 364 
9.979 405 
9.979 446 
9.979 487 
9.979 528 
9.979 569 
9.979 610 
9.979 651 
9.979 691 
9.979 732 
9.979 772 
9.979 813 
9.979 853 
9.979 893 
9.979 934 

9.979 974 

9.980 014 
9.980 054 
9.980 094 
9.980 133 
9.980 173 
9.980 213 
9.980 252 
9.980 292 
9.980 331 
9.980 370 
9.980 410 
9.980 449 
9.980 488 
9.880 527 



APPENDIX 



323 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

{Continued) 



log cosh x 



log sinh x 



log tanh x 



1.900 
1.901 
1.902 
1.903 
1.904 
1.905 
1.906 
1.907 
1.908 
1.909 
1.910 
1.911 
1.912 
1.913 
1.914 
1.915 
1.916 
1.917 
1.918 
1.919 
1.920 
1.921 
1.922 
1.923 
1.924 
1.925 
1.926 
1.927 
1.928 
1.929 
1.930 
1.931 
1.932 
1.933 
1.934 
1.935 
1.936 
1.937 
1.938 
1.939 
1.940 
1.941 
1.942 
1.943 
1.944 
1.945 
1.946 
1.947 
1.948 
1.949 



0.533 738 
0.534 153 
0.534 569 
0.534 984 
0.535 399 
0.535 815 
0.536 230 
0.536 646 
0.537 061 
0.537 477 
0.537 893 
0.538 308 
0.538 724 
0.539 140 
0.539 556 
0.539 971 
0.540 387 
0.540 803 
0.541 219 
0.541 635 
0.542 051 
0.542 467 
0.542 883 
0.543 299 
0.543 715 
0.544 132 
0.544 548 
0.544 964 
0.545 380 
0.545 797 
0.546 213 
0.546 629 
0.547 046 
0.547 462 
0.547 879 
0.548 295 
0.548 712 
0.549 128 
0.549 545 
0.549 962 
0.550 378 
0.550 795 
0.551 212 
0.551 629 
0.552 046 
0.552 463 
0.552 879 
0.553 296 
0.553 713 
0.554 130 



415 
416 
415 
415 
416 
415 
416 
415 
416 
416 
415 
416 
416 
416 
415 
416 
416 
416 
416 
416 
416 
416 
416 
416 
417 
416 
416 
416 
417 
416 
416 
417 
416 
417 
416 
417 
416 
417 
417 
416 
417 
417 
417 
417 
417 
416 
417 
417 
417 
417 



0.514 304 
0.514 758 
0.515 212 
0.515 666 
0.516 120 
0.516 574 
0.517 028 
0.517 482 
0.517 936 
0.518 390 
0.518 843 
0.519 297 
0.519 751 
0.520 205 
0.520 658 
0.521 112 
0.521565 
0.522 019 
0.522 472 
0.522 926 
0.523 379 
0.523 832 
0.524 286 
0.524 739 
0.525 192 
0.525 645 
0.526 099 
0.526 552 
0.527 005 
0.527 458 
0.527 911 
0.528 364 
0.528 817 
0.529 270 
0.529 723 
0.530 175 
0.530 628 
0.531081 
0.531 534 
0.531 986 
0.532 439 
0.532 891 
0.533 344 
0.533 797 
0.534 249 
0.0534 71 
0.535 154 
0.535 606 
0.536 059 
0.536 511 



454 
454 
454 
454 
454 
454 
454 
454 
454 
453 
454 
454 
454 
453 
454 
453 
454 
453 
454 
453 
453 
454 
453 
453 
453 
454 
453 
453 
453 
453 
453 
453 
453 
453 
452 
453 
453 
453 
452 
453 
452 
453 
453 
452 
452 
453 
452 
453 
452 
452 



9.980 566 
9.980 605 
9.980 643 
9.980 682 
9.980 721 
9.980 759 
9.980 798 
9.980 836 
9.980 874 
9.980 913 
9.980 951 

9.980 989 

9.981 027 
9.981065 
9.981 103 
9.981 140 
9.981 178 
9.981 216 
9.981253 
9.981 291 
9.981328 
9.981 365 
9.981 403 
9.981 440 
9.981 477 
9.981 514 
9.981 551 
9.981 588 
9.981 624 
9.981 661 
9.981698 
9.981 734 
9.981 771 
9.981 807 
9.981 844 
9.981 880 
9.981 916 

9.981 952 
9.981988 

9.982 024 
9.982 060 
9.982 096 
9.982 132 
9.982 168 
9.982 203 
9.982 239 
9.982 274 
9.982 310 
9.982 345 
9.982 380 



39 
38 
39 
39 
38 
39 
38 
38 
39 
38 
38 
38 
38 
38 
37 
38 
38 
37 
38 
37 
37 
38 
37 
37 
37 
37 
37 
37 
37 
37 
36 
37 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 
36 
35 
36 
35 
35 
36 



324 



APPENDIX 



TABLES OF LOGARITHMS OF HYPERBOLIC FUNCTIONS 

(Concluded) 



log cosh z 



log sinh x 



log tanh x 



1.950 
1.951 
1.952 
1.953 
1.954 
1.955 
1.956 
1.957 
1.958 
1.959 
1.960 
1.961 
1.962 
1.963 
1.964 
1.965 
1.966 
1.967 
1.968 
1.969 
1.970 
1.971 
1.972 
1.973 
1.974 
1.975 
1.976 
1.977 
1.978 
1.979 
1.980 
1.981 
1.982 
1.983 
1.984 
1.985 
1.986 
1.987 
1.988 
1.989 
1.990 
1.991 
1.992 
1.993 
1.994 
1.995 
1.996 
1.997 
1.998 
1.999 
2.000 



0.554 547 
0.554 964 
0.555 382 
0.555 799 
0.556 216 
0.556 633 
0.557 050 
0.557 468 
0.557 885 
0.558 302 
0.558 720 
0.559 137 
0.559 555 
0.559 972 
0.560 390 
0.560 807 
0.561 225 
0.561 642 
0.562 060 
0.562 478 
0.562 895 
0.563 313 
0.563 731 
0.564 149 
0.564 566 
0.564 984 
0.565 402 
0.565 820 
0.566 238 
0.566 656 
0.567 074 
0.567 492 
0.567 910 
0.568 328 
0.568 747 
0.569 165 
0.569 583 
0.570 001 
0.570 420 
0.570 838 
0.571 256 
0.571 675 
0.572 093 
0.572 511 
0.572 930 
0.573 348 
0.573 767 
0.574 186 
0.574 604 
0.575 023 
0.575 441 



417 
418 
417 
417 
417 
417 
418 
417 
417 
418 
417 
418 
417 
418 
417 
418 
417 
418 
418 
417 
418 
418 
418 
417 
418 
418 
418 
418 
418 
418 
418 
418 
418 
419 
418 
418 
418 
419 
418 
418 
419 
418 
418 
419 
418 
419 
419 
418 
419 
418 



0.536 963 
0.537 415 
0.537 868 
0.538 320 
0.538 772 
0.539 224 
0.539 676 
0.540 128 
0.540 580 
0.541 032 
0.541 484 
0.541 936 
0.542 387 
0.542 839 
0.543 291 
0.543 743 
0.544 194 
0.544 646 
0.545 098 
0.545 549 
0.546 001 
0.546 452 
0.546 904 
0.547 355 
0.547 806 
0.548 258 
0.548 709 
0.549 160 
0.549 612 
0.550 063 
0.550 514 
0.550 965 
0.551 416 
0.551 868 
0.552 319 
0.552 770 
-0.553 221 
0.553 672 
0.554 122 
0.554 573 
0.555 024 
0.555 475 
0.555 926 
0.556 377 
0.556 827 
0.557 278 
0.557 729 
0.558 179 
0.558 630 
0.559 080 
0.559 531 



452 
453 
452 
452 
452 
452 
452 
452 
452 
452 
452 
451 
452 
452 
452 
451 
452 
452 
451 
452 

451 
452 
451 
451 
452 
451 
451 
452 
451 
451 
451 
451 
452 
451 
451 
451 
451 
450 
451 
451 
451 
451 
451 
450 
451 
451 
450 
451 
450 
451 



9.982 416 
9.982 451 
9.982 486 
9.982 521 
9.982 556 
9.982 591 
9.982 626 
9.982 660 
9.982 695 
9.982 729 
9.982 764 
9.982 798 
9.982 833 
9.982 867 
9.982 901 
9.982 936 

9.982 970 

9.983 004 
9.983 038 
9.983 072 
9.983 105 
9.983 139 
9.983 173 
9.983 206 
9.983 240 
9.983 274 
9.983 307 
9.983 340 
9.983 374 
9.983 407 
9.983 440 
9.983 473 
9.983 506 
9.983 539 
9.983 572 
9.983 605 
9.983 638 
9.983 670 
9.983 703 
9.983 735 
9.983 768 
9.983 800 
9.983 833 
9.983 865 
9.983 897 
9.983 930 
9.983 962 

9.983 994 

9.984 026 
9.984 058 
9.984 089 



35 
35 
35 
35 
35 
35 
34 
35 
34 
35 
34 
35 
34 
34 
35 
34 
34 
34 
34 
33 
34 
34 
33 
34 
34 
33 
33 
34 
33 
33 
33 
33 
33 
33 
33 
33 
32 
33 
32 
33 
32 
33 
32 
32 
33 
32 
32 
32 
32 
31 



APPENDIX 



325 



Curve No. 
FORM A 
Separation of Odd and Even Harmonics for 72 Ordinate Analysis 



Vm' — 5 (Vm — ym+ii). 



y» 



h (ym + y m +3t,). 





Data 




Diff. 


Sum. 




Data 




Diff. 


Sum. 


tn 


Vm 


Vm 




2Vm' 


2»„" 




Vm 


Vm 




2Vm 


2Vm" 









36 






18 






54 






l 






37 






19 






55 






2 






38 






20 






56 






3 






39 






21 






57 






4 






40 






22 






58 






5 






41 






23 






59 






6 






42 






24 






60 






7 






43 






25 






61 






8 






44 






26 






62 






9 






45 






27 






63 






10 






46 






28 






64 






11 






47 






29 






65 






12 






48 






30 






66 






13 






49 






31 






67 






14 






50 






32 






68 






15 






51 






33 






69 






16 






52 






34 






70 






17 






53 






35 






71 







Use y m ' in analysis for odd harmonics. Carry y m " to Form E as data. 
Remarks. 



Computed by 



326 



AIM'KNDIX 



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APPENDIX 



327 



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328 



APPENDIX 



Curve No. 
FORM E 
Separation of Odd and Even Harmonics for 36 Ordinate Analysis 

Vm' = \ (Vm — ym+u). J/m" = Hj/"» + Vm+is). 





Data 




Diff. 


Sum. 


TFt 


*m 


9m 




2»«' 


2v m " 









18 






1 






19 






2 






20 






3 






21 






4 






22 






5 






23 






6 






24 






7 






25 






8 






26 






9 






27 






10 






28 






11 






29 






12 






30 






13 






31 






14 






32 






15 






33 






16 






34 






17 






35 







Use y m ' in analysis for odd harmonics. Carry y m " to Form H as data. 
Remarks. 



Computed by 



APPENDIX 



329 



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330 



AIM'KNDIX 



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APPENDIX 



331 



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332 



APPENDIX 



Curve No. 



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